Random Quadratic Equation Generator

A quadratic equation to solve, with the worked answer covered until you want it.

x² − 8x + 15 = 0

Next step Draw a fraction to simplify Draw a decimal number

How it works

Give me a random quadratic equation: press the button and one appears in standard form, such as x² − 7x + 12 = 0. Solve it on paper, then press Show answer. Every answer was checked by putting it back into its equation.

Read the full guide

Choose the kind you are practicing

After the first press a dropdown sorts the equations by method. Factoring, x² first gives equations such as x² − 3x − 10 = 0, which split into two brackets with whole-number roots. Factoring, 2x² to 6x² needs brackets such as (x + 3)(2x − 1), so a root is often a fraction. Quadratic formula equations do not factor: the answer is exact, such as 4 ± 2√2, with decimals for checking a calculator. One repeated root equations are perfect squares. No real roots answers show b² − 4ac coming out below zero.

Make a practice set

Press once for each random quadratic equation you want, copy it out before the next press, and leave every answer covered until the set is done.

Questions and answers

What is a quadratic equation in standard form?

ax² + bx + c = 0, where a is not zero. Every equation here is written that way, with whole-number coefficients that share no common factor.

How do I know which method to use?

Work out b² − 4ac. A perfect square such as 25 means the equation factors. Zero means one repeated root. Any other positive number needs the quadratic formula, and a negative number means no real roots.

How were the answers checked?

A script puts each root back into its equation with exact arithmetic and keeps the equation only if the result is 0. Roots with a square root in them are checked in exact form, not as rounded decimals.

All 300 equations

Factoring, x² first 100

x² + 10x + 16 = 0 (x + 8)(x + 2) = 0, so x = −8 or x = −2.
x² + 10x + 21 = 0 (x + 7)(x + 3) = 0, so x = −7 or x = −3.
x² + 10x + 24 = 0 (x + 6)(x + 4) = 0, so x = −6 or x = −4.
x² + 10x + 9 = 0 (x + 9)(x + 1) = 0, so x = −9 or x = −1.
x² + 11x + 18 = 0 (x + 9)(x + 2) = 0, so x = −9 or x = −2.
x² + 11x + 24 = 0 (x + 8)(x + 3) = 0, so x = −8 or x = −3.
x² + 11x + 28 = 0 (x + 7)(x + 4) = 0, so x = −7 or x = −4.
x² + 11x + 30 = 0 (x + 6)(x + 5) = 0, so x = −6 or x = −5.
x² + 12x + 27 = 0 (x + 9)(x + 3) = 0, so x = −9 or x = −3.
x² + 12x + 35 = 0 (x + 7)(x + 5) = 0, so x = −7 or x = −5.
x² + 2x − 24 = 0 (x + 6)(x − 4) = 0, so x = −6 or x = 4.
x² + 2x − 3 = 0 (x + 3)(x − 1) = 0, so x = −3 or x = 1.
x² + 2x − 35 = 0 (x + 7)(x − 5) = 0, so x = −7 or x = 5.
x² + 3x + 2 = 0 (x + 2)(x + 1) = 0, so x = −2 or x = −1.

Show all 300 equations (286 more)

Factoring, x² first, continued

x² + 3x − 10 = 0 (x + 5)(x − 2) = 0, so x = −5 or x = 2.
x² + 3x − 28 = 0 (x + 7)(x − 4) = 0, so x = −7 or x = 4.
x² + 3x − 4 = 0 (x + 4)(x − 1) = 0, so x = −4 or x = 1.
x² + 3x − 40 = 0 (x + 8)(x − 5) = 0, so x = −8 or x = 5.
x² + 4x + 3 = 0 (x + 3)(x + 1) = 0, so x = −3 or x = −1.
x² + 4x − 32 = 0 (x + 8)(x − 4) = 0, so x = −8 or x = 4.
x² + 4x − 5 = 0 (x + 5)(x − 1) = 0, so x = −5 or x = 1.
x² + 5x + 6 = 0 (x + 3)(x + 2) = 0, so x = −3 or x = −2.
x² + 5x = 0 x(x + 5) = 0, so x = 0 or x = −5.
x² + 5x − 14 = 0 (x + 7)(x − 2) = 0, so x = −7 or x = 2.
x² + 5x − 24 = 0 (x + 8)(x − 3) = 0, so x = −8 or x = 3.
x² + 5x − 36 = 0 (x + 9)(x − 4) = 0, so x = −9 or x = 4.
x² + 5x − 6 = 0 (x + 6)(x − 1) = 0, so x = −6 or x = 1.
x² + 6x + 8 = 0 (x + 4)(x + 2) = 0, so x = −4 or x = −2.
x² + 6x = 0 x(x + 6) = 0, so x = 0 or x = −6.
x² + 6x − 16 = 0 (x + 8)(x − 2) = 0, so x = −8 or x = 2.
x² + 6x − 27 = 0 (x + 9)(x − 3) = 0, so x = −9 or x = 3.
x² + 6x − 7 = 0 (x + 7)(x − 1) = 0, so x = −7 or x = 1.
x² + 7x + 10 = 0 (x + 5)(x + 2) = 0, so x = −5 or x = −2.
x² + 7x + 12 = 0 (x + 4)(x + 3) = 0, so x = −4 or x = −3.
x² + 7x + 6 = 0 (x + 6)(x + 1) = 0, so x = −6 or x = −1.
x² + 7x = 0 x(x + 7) = 0, so x = 0 or x = −7.
x² + 7x − 8 = 0 (x + 8)(x − 1) = 0, so x = −8 or x = 1.
x² + 8x + 15 = 0 (x + 5)(x + 3) = 0, so x = −5 or x = −3.
x² + 8x + 7 = 0 (x + 7)(x + 1) = 0, so x = −7 or x = −1.
x² + 8x − 9 = 0 (x + 9)(x − 1) = 0, so x = −9 or x = 1.
x² + 9x + 18 = 0 (x + 6)(x + 3) = 0, so x = −6 or x = −3.
x² + 9x + 20 = 0 (x + 5)(x + 4) = 0, so x = −5 or x = −4.
x² + x − 12 = 0 (x + 4)(x − 3) = 0, so x = −4 or x = 3.
x² + x − 20 = 0 (x + 5)(x − 4) = 0, so x = −5 or x = 4.
x² + x − 30 = 0 (x + 6)(x − 5) = 0, so x = −6 or x = 5.
x² + x − 6 = 0 (x + 3)(x − 2) = 0, so x = −3 or x = 2.
x² − 10x + 21 = 0 (x − 3)(x − 7) = 0, so x = 3 or x = 7.
x² − 10x + 24 = 0 (x − 4)(x − 6) = 0, so x = 4 or x = 6.
x² − 10x + 9 = 0 (x − 1)(x − 9) = 0, so x = 1 or x = 9.
x² − 11x + 24 = 0 (x − 3)(x − 8) = 0, so x = 3 or x = 8.
x² − 11x + 28 = 0 (x − 4)(x − 7) = 0, so x = 4 or x = 7.
x² − 11x + 30 = 0 (x − 5)(x − 6) = 0, so x = 5 or x = 6.
x² − 12x + 32 = 0 (x − 4)(x − 8) = 0, so x = 4 or x = 8.
x² − 12x + 35 = 0 (x − 5)(x − 7) = 0, so x = 5 or x = 7.
x² − 13x + 36 = 0 (x − 4)(x − 9) = 0, so x = 4 or x = 9.
x² − 13x + 40 = 0 (x − 5)(x − 8) = 0, so x = 5 or x = 8.
x² − 16 = 0 (x + 4)(x − 4) = 0, so x = −4 or x = 4.
x² − 25 = 0 (x + 5)(x − 5) = 0, so x = −5 or x = 5.
x² − 2x − 3 = 0 (x + 1)(x − 3) = 0, so x = −1 or x = 3.
x² − 2x − 35 = 0 (x + 5)(x − 7) = 0, so x = −5 or x = 7.
x² − 2x − 8 = 0 (x + 2)(x − 4) = 0, so x = −2 or x = 4.
x² − 36 = 0 (x + 6)(x − 6) = 0, so x = −6 or x = 6.
x² − 3x + 2 = 0 (x − 1)(x − 2) = 0, so x = 1 or x = 2.
x² − 3x = 0 x(x − 3) = 0, so x = 0 or x = 3.
x² − 3x − 10 = 0 (x + 2)(x − 5) = 0, so x = −2 or x = 5.
x² − 3x − 18 = 0 (x + 3)(x − 6) = 0, so x = −3 or x = 6.
x² − 3x − 28 = 0 (x + 4)(x − 7) = 0, so x = −4 or x = 7.
x² − 3x − 4 = 0 (x + 1)(x − 4) = 0, so x = −1 or x = 4.
x² − 4x + 3 = 0 (x − 1)(x − 3) = 0, so x = 1 or x = 3.
x² − 4x = 0 x(x − 4) = 0, so x = 0 or x = 4.
x² − 4x − 12 = 0 (x + 2)(x − 6) = 0, so x = −2 or x = 6.
x² − 4x − 32 = 0 (x + 4)(x − 8) = 0, so x = −4 or x = 8.
x² − 4x − 5 = 0 (x + 1)(x − 5) = 0, so x = −1 or x = 5.
x² − 5x + 4 = 0 (x − 1)(x − 4) = 0, so x = 1 or x = 4.
x² − 5x − 14 = 0 (x + 2)(x − 7) = 0, so x = −2 or x = 7.
x² − 5x − 36 = 0 (x + 4)(x − 9) = 0, so x = −4 or x = 9.
x² − 5x − 6 = 0 (x + 1)(x − 6) = 0, so x = −1 or x = 6.
x² − 6x + 5 = 0 (x − 1)(x − 5) = 0, so x = 1 or x = 5.
x² − 6x + 8 = 0 (x − 2)(x − 4) = 0, so x = 2 or x = 4.
x² − 6x − 16 = 0 (x + 2)(x − 8) = 0, so x = −2 or x = 8.
x² − 6x − 7 = 0 (x + 1)(x − 7) = 0, so x = −1 or x = 7.
x² − 7x + 10 = 0 (x − 2)(x − 5) = 0, so x = 2 or x = 5.
x² − 7x + 12 = 0 (x − 3)(x − 4) = 0, so x = 3 or x = 4.
x² − 7x + 6 = 0 (x − 1)(x − 6) = 0, so x = 1 or x = 6.
x² − 7x = 0 x(x − 7) = 0, so x = 0 or x = 7.
x² − 7x − 18 = 0 (x + 2)(x − 9) = 0, so x = −2 or x = 9.
x² − 7x − 8 = 0 (x + 1)(x − 8) = 0, so x = −1 or x = 8.
x² − 8x + 12 = 0 (x − 2)(x − 6) = 0, so x = 2 or x = 6.
x² − 8x + 15 = 0 (x − 3)(x − 5) = 0, so x = 3 or x = 5.
x² − 8x + 7 = 0 (x − 1)(x − 7) = 0, so x = 1 or x = 7.
x² − 9 = 0 (x + 3)(x − 3) = 0, so x = −3 or x = 3.
x² − 9x + 14 = 0 (x − 2)(x − 7) = 0, so x = 2 or x = 7.
x² − 9x + 18 = 0 (x − 3)(x − 6) = 0, so x = 3 or x = 6.
x² − 9x + 20 = 0 (x − 4)(x − 5) = 0, so x = 4 or x = 5.
x² − 9x + 8 = 0 (x − 1)(x − 8) = 0, so x = 1 or x = 8.
x² − 9x = 0 x(x − 9) = 0, so x = 0 or x = 9.
x² − x = 0 x(x − 1) = 0, so x = 0 or x = 1.
x² − x − 20 = 0 (x + 4)(x − 5) = 0, so x = −4 or x = 5.
x² − x − 30 = 0 (x + 5)(x − 6) = 0, so x = −5 or x = 6.
x² − x − 6 = 0 (x + 2)(x − 3) = 0, so x = −2 or x = 3.

Factoring, 2x² to 6x² 80

2x² + 11x − 6 = 0 (x + 6)(2x − 1) = 0, so x = −6 or x = 1/2.
2x² + 13x + 15 = 0 (x + 5)(2x + 3) = 0, so x = −5 or x = −3/2.
2x² + 13x + 20 = 0 (x + 4)(2x + 5) = 0, so x = −4 or x = −5/2.
2x² + 13x + 21 = 0 (2x + 7)(x + 3) = 0, so x = −7/2 or x = −3.
2x² + 17x + 21 = 0 (x + 7)(2x + 3) = 0, so x = −7 or x = −3/2.
2x² + 3x + 1 = 0 (x + 1)(2x + 1) = 0, so x = −1 or x = −1/2.
2x² + 5x + 2 = 0 (x + 2)(2x + 1) = 0, so x = −2 or x = −1/2.
2x² + 5x + 3 = 0 (2x + 3)(x + 1) = 0, so x = −3/2 or x = −1.
2x² + 5x − 3 = 0 (x + 3)(2x − 1) = 0, so x = −3 or x = 1/2.
2x² + 7x − 15 = 0 (x + 5)(2x − 3) = 0, so x = −5 or x = 3/2.
2x² + 7x − 4 = 0 (x + 4)(2x − 1) = 0, so x = −4 or x = 1/2.
2x² + 9x + 7 = 0 (2x + 7)(x + 1) = 0, so x = −7/2 or x = −1.
2x² + 9x − 5 = 0 (x + 5)(2x − 1) = 0, so x = −5 or x = 1/2.
2x² + x − 1 = 0 (x + 1)(2x − 1) = 0, so x = −1 or x = 1/2.
2x² + x − 15 = 0 (x + 3)(2x − 5) = 0, so x = −3 or x = 5/2.
2x² − 11x − 21 = 0 (2x + 3)(x − 7) = 0, so x = −3/2 or x = 7.
2x² − 13x − 7 = 0 (2x + 1)(x − 7) = 0, so x = −1/2 or x = 7.
2x² − 3x − 14 = 0 (x + 2)(2x − 7) = 0, so x = −2 or x = 7/2.
2x² − 3x − 2 = 0 (2x + 1)(x − 2) = 0, so x = −1/2 or x = 2.
2x² − 3x − 20 = 0 (2x + 5)(x − 4) = 0, so x = −5/2 or x = 4.
2x² − 5x − 12 = 0 (2x + 3)(x − 4) = 0, so x = −3/2 or x = 4.
2x² − 7x + 5 = 0 (x − 1)(2x − 5) = 0, so x = 1 or x = 5/2.
2x² − 7x + 6 = 0 (2x − 3)(x − 2) = 0, so x = 3/2 or x = 2.
2x² − 9x + 10 = 0 (x − 2)(2x − 5) = 0, so x = 2 or x = 5/2.
2x² − 9x + 4 = 0 (2x − 1)(x − 4) = 0, so x = 1/2 or x = 4.
2x² − x − 10 = 0 (x + 2)(2x − 5) = 0, so x = −2 or x = 5/2.
3x² + 10x + 3 = 0 (x + 3)(3x + 1) = 0, so x = −3 or x = −1/3.
3x² + 11x + 6 = 0 (x + 3)(3x + 2) = 0, so x = −3 or x = −2/3.
3x² + 13x + 14 = 0 (3x + 7)(x + 2) = 0, so x = −7/3 or x = −2.
3x² + 17x − 6 = 0 (x + 6)(3x − 1) = 0, so x = −6 or x = 1/3.
3x² + 19x + 6 = 0 (x + 6)(3x + 1) = 0, so x = −6 or x = −1/3.
3x² + 20x − 7 = 0 (x + 7)(3x − 1) = 0, so x = −7 or x = 1/3.
3x² + 2x − 21 = 0 (x + 3)(3x − 7) = 0, so x = −3 or x = 7/3.
3x² + 2x − 5 = 0 (3x + 5)(x − 1) = 0, so x = −5/3 or x = 1.
3x² + 4x + 1 = 0 (x + 1)(3x + 1) = 0, so x = −1 or x = −1/3.
3x² + 4x − 15 = 0 (x + 3)(3x − 5) = 0, so x = −3 or x = 5/3.
3x² + 5x − 2 = 0 (x + 2)(3x − 1) = 0, so x = −2 or x = 1/3.
3x² + 7x + 2 = 0 (x + 2)(3x + 1) = 0, so x = −2 or x = −1/3.
3x² + 7x − 6 = 0 (x + 3)(3x − 2) = 0, so x = −3 or x = 2/3.
3x² + 8x + 4 = 0 (x + 2)(3x + 2) = 0, so x = −2 or x = −2/3.
3x² + 8x + 5 = 0 (3x + 5)(x + 1) = 0, so x = −5/3 or x = −1.
3x² − 13x + 12 = 0 (3x − 4)(x − 3) = 0, so x = 4/3 or x = 3.
3x² − 14x − 24 = 0 (3x + 4)(x − 6) = 0, so x = −4/3 or x = 6.
3x² − 16x − 12 = 0 (3x + 2)(x − 6) = 0, so x = −2/3 or x = 6.
3x² − 17x + 10 = 0 (3x − 2)(x − 5) = 0, so x = 2/3 or x = 5.
3x² − 19x − 14 = 0 (3x + 2)(x − 7) = 0, so x = −2/3 or x = 7.
3x² − 4x − 4 = 0 (3x + 2)(x − 2) = 0, so x = −2/3 or x = 2.
3x² − 7x − 20 = 0 (3x + 5)(x − 4) = 0, so x = −5/3 or x = 4.
4x² + 13x + 10 = 0 (x + 2)(4x + 5) = 0, so x = −2 or x = −5/4.
4x² + 17x − 15 = 0 (x + 5)(4x − 3) = 0, so x = −5 or x = 3/4.
4x² + 20x + 21 = 0 (2x + 7)(2x + 3) = 0, so x = −7/2 or x = −3/2.
4x² + 3x − 10 = 0 (x + 2)(4x − 5) = 0, so x = −2 or x = 5/4.
4x² + 5x − 21 = 0 (x + 3)(4x − 7) = 0, so x = −3 or x = 7/4.
4x² + 7x − 2 = 0 (x + 2)(4x − 1) = 0, so x = −2 or x = 1/4.
4x² + 9x − 9 = 0 (x + 3)(4x − 3) = 0, so x = −3 or x = 3/4.
4x² + x − 5 = 0 (4x + 5)(x − 1) = 0, so x = −5/4 or x = 1.
4x² − 13x + 3 = 0 (4x − 1)(x − 3) = 0, so x = 1/4 or x = 3.
4x² − 23x + 15 = 0 (4x − 3)(x − 5) = 0, so x = 3/4 or x = 5.
4x² − 3x − 10 = 0 (4x + 5)(x − 2) = 0, so x = −5/4 or x = 2.
4x² − 8x − 5 = 0 (2x + 1)(2x − 5) = 0, so x = −1/2 or x = 5/2.
5x² + 13x − 6 = 0 (x + 3)(5x − 2) = 0, so x = −3 or x = 2/5.
5x² + 17x + 6 = 0 (x + 3)(5x + 2) = 0, so x = −3 or x = −2/5.
5x² + 2x − 7 = 0 (5x + 7)(x − 1) = 0, so x = −7/5 or x = 1.
5x² + 8x − 21 = 0 (x + 3)(5x − 7) = 0, so x = −3 or x = 7/5.
5x² + 8x − 4 = 0 (x + 2)(5x − 2) = 0, so x = −2 or x = 2/5.
5x² − 13x + 6 = 0 (5x − 3)(x − 2) = 0, so x = 3/5 or x = 2.
5x² − 19x − 4 = 0 (5x + 1)(x − 4) = 0, so x = −1/5 or x = 4.
5x² − 21x − 20 = 0 (5x + 4)(x − 5) = 0, so x = −4/5 or x = 5.
5x² − 24x − 5 = 0 (5x + 1)(x − 5) = 0, so x = −1/5 or x = 5.
5x² − 3x − 2 = 0 (5x + 2)(x − 1) = 0, so x = −2/5 or x = 1.
6x² + 11x − 2 = 0 (x + 2)(6x − 1) = 0, so x = −2 or x = 1/6.
6x² + 11x − 21 = 0 (x + 3)(6x − 7) = 0, so x = −3 or x = 7/6.
6x² + 13x + 6 = 0 (2x + 3)(3x + 2) = 0, so x = −3/2 or x = −2/3.
6x² + 17x + 7 = 0 (3x + 7)(2x + 1) = 0, so x = −7/3 or x = −1/2.
6x² + 17x − 3 = 0 (x + 3)(6x − 1) = 0, so x = −3 or x = 1/6.
6x² + 23x + 15 = 0 (x + 3)(6x + 5) = 0, so x = −3 or x = −5/6.
6x² + 23x + 7 = 0 (2x + 7)(3x + 1) = 0, so x = −7/2 or x = −1/3.
6x² + x − 2 = 0 (3x + 2)(2x − 1) = 0, so x = −2/3 or x = 1/2.
6x² − 11x + 5 = 0 (6x − 5)(x − 1) = 0, so x = 5/6 or x = 1.
6x² − 23x + 21 = 0 (2x − 3)(3x − 7) = 0, so x = 3/2 or x = 7/3.

Quadratic formula 60

2x² + 3x − 3 = 0 By the quadratic formula, x = (−3 ± √33)/4: about −2.19 and 0.69.
2x² + 3x − 7 = 0 By the quadratic formula, x = (−3 ± √65)/4: about −2.77 and 1.27.
2x² + 5x + 1 = 0 By the quadratic formula, x = (−5 ± √17)/4: about −2.28 and −0.22.
2x² + 5x − 8 = 0 By the quadratic formula, x = (−5 ± √89)/4: about −3.61 and 1.11.
2x² + 6x − 3 = 0 By the quadratic formula, x = (−3 ± √15)/2: about −3.44 and 0.44.
2x² + 7x − 1 = 0 By the quadratic formula, x = (−7 ± √57)/4: about −3.64 and 0.14.
2x² + 7x − 7 = 0 By the quadratic formula, x = (−7 ± √105)/4: about −4.31 and 0.81.
2x² + 8x + 5 = 0 By the quadratic formula, x = (−4 ± √6)/2: about −3.22 and −0.78.
2x² + 8x − 7 = 0 By the quadratic formula, x = (−4 ± √30)/2: about −4.74 and 0.74.
2x² − 2x − 1 = 0 By the quadratic formula, x = (1 ± √3)/2: about −0.37 and 1.37.
2x² − 3x − 6 = 0 By the quadratic formula, x = (3 ± √57)/4: about −1.14 and 2.64.
2x² − 3x − 8 = 0 By the quadratic formula, x = (3 ± √73)/4: about −1.39 and 2.89.
2x² − 4x − 1 = 0 By the quadratic formula, x = (2 ± √6)/2: about −0.22 and 2.22.
2x² − 4x − 7 = 0 By the quadratic formula, x = (2 ± 3√2)/2: about −1.12 and 3.12.
2x² − 5x − 4 = 0 By the quadratic formula, x = (5 ± √57)/4: about −0.64 and 3.14.
2x² − 5x − 9 = 0 By the quadratic formula, x = (5 ± √97)/4: about −1.21 and 3.71.
2x² − 6x − 9 = 0 By the quadratic formula, x = (3 ± 3√3)/2: about −1.10 and 4.10.
2x² − 7x − 7 = 0 By the quadratic formula, x = (7 ± √105)/4: about −0.81 and 4.31.
2x² − 8x + 1 = 0 By the quadratic formula, x = (4 ± √14)/2: about 0.13 and 3.87.
2x² − x − 5 = 0 By the quadratic formula, x = (1 ± √41)/4: about −1.35 and 1.85.
x² + 3x + 1 = 0 By the quadratic formula, x = (−3 ± √5)/2: about −2.62 and −0.38.
x² + 3x − 1 = 0 By the quadratic formula, x = (−3 ± √13)/2: about −3.30 and 0.30.
x² + 3x − 5 = 0 By the quadratic formula, x = (−3 ± √29)/2: about −4.19 and 1.19.
x² + 3x − 7 = 0 By the quadratic formula, x = (−3 ± √37)/2: about −4.54 and 1.54.
x² + 5x − 1 = 0 By the quadratic formula, x = (−5 ± √29)/2: about −5.19 and 0.19.
x² + 5x − 5 = 0 By the quadratic formula, x = (−5 ± 3√5)/2: about −5.85 and 0.85.
x² + 5x − 8 = 0 By the quadratic formula, x = (−5 ± √57)/2: about −6.27 and 1.27.
x² + 5x − 9 = 0 By the quadratic formula, x = (−5 ± √61)/2: about −6.41 and 1.41.
x² + 6x + 3 = 0 By the quadratic formula, x = −3 ± √6: about −5.45 and −0.55.
x² + 6x + 7 = 0 By the quadratic formula, x = −3 ± √2: about −4.41 and −1.59.
x² + 6x − 5 = 0 By the quadratic formula, x = −3 ± √14: about −6.74 and 0.74.
x² + 7x + 2 = 0 By the quadratic formula, x = (−7 ± √41)/2: about −6.70 and −0.30.
x² + 7x + 8 = 0 By the quadratic formula, x = (−7 ± √17)/2: about −5.56 and −1.44.
x² + 7x − 4 = 0 By the quadratic formula, x = (−7 ± √65)/2: about −7.53 and 0.53.
x² + 7x − 6 = 0 By the quadratic formula, x = (−7 ± √73)/2: about −7.77 and 0.77.
x² + 7x − 7 = 0 By the quadratic formula, x = (−7 ± √77)/2: about −7.89 and 0.89.
x² + 8x + 8 = 0 By the quadratic formula, x = −4 ± 2√2: about −6.83 and −1.17.
x² + 8x − 3 = 0 By the quadratic formula, x = −4 ± √19: about −8.36 and 0.36.
x² + x − 1 = 0 By the quadratic formula, x = (−1 ± √5)/2: about −1.62 and 0.62.
x² + x − 3 = 0 By the quadratic formula, x = (−1 ± √13)/2: about −2.30 and 1.30.
x² − 2x − 5 = 0 By the quadratic formula, x = 1 ± √6: about −1.45 and 3.45.
x² − 2x − 6 = 0 By the quadratic formula, x = 1 ± √7: about −1.65 and 3.65.
x² − 3x − 6 = 0 By the quadratic formula, x = (3 ± √33)/2: about −1.37 and 4.37.
x² − 4x − 1 = 0 By the quadratic formula, x = 2 ± √5: about −0.24 and 4.24.
x² − 4x − 7 = 0 By the quadratic formula, x = 2 ± √11: about −1.32 and 5.32.
x² − 5x − 4 = 0 By the quadratic formula, x = (5 ± √41)/2: about −0.70 and 5.70.
x² − 5x − 8 = 0 By the quadratic formula, x = (5 ± √57)/2: about −1.27 and 6.27.
x² − 6x + 4 = 0 By the quadratic formula, x = 3 ± √5: about 0.76 and 5.24.
x² − 6x + 6 = 0 By the quadratic formula, x = 3 ± √3: about 1.27 and 4.73.
x² − 6x − 2 = 0 By the quadratic formula, x = 3 ± √11: about −0.32 and 6.32.
x² − 6x − 3 = 0 By the quadratic formula, x = 3 ± 2√3: about −0.46 and 6.46.
x² − 6x − 4 = 0 By the quadratic formula, x = 3 ± √13: about −0.61 and 6.61.
x² − 7x + 3 = 0 By the quadratic formula, x = (7 ± √37)/2: about 0.46 and 6.54.
x² − 7x + 4 = 0 By the quadratic formula, x = (7 ± √33)/2: about 0.63 and 6.37.
x² − 7x − 1 = 0 By the quadratic formula, x = (7 ± √53)/2: about −0.14 and 7.14.
x² − 7x − 3 = 0 By the quadratic formula, x = (7 ± √61)/2: about −0.41 and 7.41.
x² − 8x + 8 = 0 By the quadratic formula, x = 4 ± 2√2: about 1.17 and 6.83.
x² − 8x − 5 = 0 By the quadratic formula, x = 4 ± √21: about −0.58 and 8.58.
x² − x − 3 = 0 By the quadratic formula, x = (1 ± √13)/2: about −1.30 and 2.30.
x² − x − 9 = 0 By the quadratic formula, x = (1 ± √37)/2: about −2.54 and 3.54.

One repeated root 30

4x² + 12x + 9 = 0 (2x + 3)² = 0, so x = −3/2, a repeated root.
4x² + 20x + 25 = 0 (2x + 5)² = 0, so x = −5/2, a repeated root.
4x² + 4x + 1 = 0 (2x + 1)² = 0, so x = −1/2, a repeated root.
4x² − 12x + 9 = 0 (2x − 3)² = 0, so x = 3/2, a repeated root.
4x² − 20x + 25 = 0 (2x − 5)² = 0, so x = 5/2, a repeated root.
4x² − 4x + 1 = 0 (2x − 1)² = 0, so x = 1/2, a repeated root.
9x² + 12x + 4 = 0 (3x + 2)² = 0, so x = −2/3, a repeated root.
9x² + 24x + 16 = 0 (3x + 4)² = 0, so x = −4/3, a repeated root.
9x² + 6x + 1 = 0 (3x + 1)² = 0, so x = −1/3, a repeated root.
9x² − 12x + 4 = 0 (3x − 2)² = 0, so x = 2/3, a repeated root.
9x² − 24x + 16 = 0 (3x − 4)² = 0, so x = 4/3, a repeated root.
9x² − 6x + 1 = 0 (3x − 1)² = 0, so x = 1/3, a repeated root.
x² + 10x + 25 = 0 (x + 5)² = 0, so x = −5, a repeated root.
x² + 12x + 36 = 0 (x + 6)² = 0, so x = −6, a repeated root.
x² + 14x + 49 = 0 (x + 7)² = 0, so x = −7, a repeated root.
x² + 16x + 64 = 0 (x + 8)² = 0, so x = −8, a repeated root.
x² + 18x + 81 = 0 (x + 9)² = 0, so x = −9, a repeated root.
x² + 2x + 1 = 0 (x + 1)² = 0, so x = −1, a repeated root.
x² + 4x + 4 = 0 (x + 2)² = 0, so x = −2, a repeated root.
x² + 6x + 9 = 0 (x + 3)² = 0, so x = −3, a repeated root.
x² + 8x + 16 = 0 (x + 4)² = 0, so x = −4, a repeated root.
x² − 10x + 25 = 0 (x − 5)² = 0, so x = 5, a repeated root.
x² − 12x + 36 = 0 (x − 6)² = 0, so x = 6, a repeated root.
x² − 14x + 49 = 0 (x − 7)² = 0, so x = 7, a repeated root.
x² − 16x + 64 = 0 (x − 8)² = 0, so x = 8, a repeated root.
x² − 18x + 81 = 0 (x − 9)² = 0, so x = 9, a repeated root.
x² − 2x + 1 = 0 (x − 1)² = 0, so x = 1, a repeated root.
x² − 4x + 4 = 0 (x − 2)² = 0, so x = 2, a repeated root.
x² − 6x + 9 = 0 (x − 3)² = 0, so x = 3, a repeated root.
x² − 8x + 16 = 0 (x − 4)² = 0, so x = 4, a repeated root.

No real roots 30

2x² + 3 = 0 b² − 4ac = 0 − 24 = −24, below zero, so there are no real roots.
2x² + 5x + 4 = 0 b² − 4ac = 25 − 32 = −7, below zero, so there are no real roots.
2x² + 5x + 8 = 0 b² − 4ac = 25 − 64 = −39, below zero, so there are no real roots.
2x² + 6x + 5 = 0 b² − 4ac = 36 − 40 = −4, below zero, so there are no real roots.
2x² − 3x + 4 = 0 b² − 4ac = 9 − 32 = −23, below zero, so there are no real roots.
2x² − 4x + 5 = 0 b² − 4ac = 16 − 40 = −24, below zero, so there are no real roots.
2x² − 4x + 9 = 0 b² − 4ac = 16 − 72 = −56, below zero, so there are no real roots.
2x² − 5x + 4 = 0 b² − 4ac = 25 − 32 = −7, below zero, so there are no real roots.
2x² − x + 5 = 0 b² − 4ac = 1 − 40 = −39, below zero, so there are no real roots.
3x² + 2x + 1 = 0 b² − 4ac = 4 − 12 = −8, below zero, so there are no real roots.
3x² + 5 = 0 b² − 4ac = 0 − 60 = −60, below zero, so there are no real roots.
3x² + 5x + 4 = 0 b² − 4ac = 25 − 48 = −23, below zero, so there are no real roots.
3x² + 6x + 5 = 0 b² − 4ac = 36 − 60 = −24, below zero, so there are no real roots.
3x² + 6x + 8 = 0 b² − 4ac = 36 − 96 = −60, below zero, so there are no real roots.
3x² − x + 1 = 0 b² − 4ac = 1 − 12 = −11, below zero, so there are no real roots.
x² + 3 = 0 b² − 4ac = 0 − 12 = −12, below zero, so there are no real roots.
x² + 4 = 0 b² − 4ac = 0 − 16 = −16, below zero, so there are no real roots.
x² + 4x + 8 = 0 b² − 4ac = 16 − 32 = −16, below zero, so there are no real roots.
x² + 6x + 11 = 0 b² − 4ac = 36 − 44 = −8, below zero, so there are no real roots.
x² + x + 3 = 0 b² − 4ac = 1 − 12 = −11, below zero, so there are no real roots.
x² − 2x + 12 = 0 b² − 4ac = 4 − 48 = −44, below zero, so there are no real roots.
x² − 3x + 3 = 0 b² − 4ac = 9 − 12 = −3, below zero, so there are no real roots.
x² − 3x + 7 = 0 b² − 4ac = 9 − 28 = −19, below zero, so there are no real roots.
x² − 4x + 6 = 0 b² − 4ac = 16 − 24 = −8, below zero, so there are no real roots.
x² − 4x + 8 = 0 b² − 4ac = 16 − 32 = −16, below zero, so there are no real roots.
x² − 4x + 9 = 0 b² − 4ac = 16 − 36 = −20, below zero, so there are no real roots.
x² − 5x + 9 = 0 b² − 4ac = 25 − 36 = −11, below zero, so there are no real roots.
x² − x + 1 = 0 b² − 4ac = 1 − 4 = −3, below zero, so there are no real roots.
x² − x + 11 = 0 b² − 4ac = 1 − 44 = −43, below zero, so there are no real roots.
x² − x + 8 = 0 b² − 4ac = 1 − 32 = −31, below zero, so there are no real roots.

How this list was compiled: Own work by script; no equation or answer is typed by hand. 300 equations in standard form, ax² + bx + c = 0, with whole-number coefficients that share no common factor. The 100 with x² first are built from two whole-number roots; the 80 that start 2x² to 6x² are built from two factors (px + q)(rx + s); the 30 with one repeated root are built as a perfect square; the 60 for the quadratic formula and the 30 with no real roots are built from their coefficients and sorted by the sign of b² − 4ac. Every answer is put back into its equation with exact arithmetic before it is written: each whole-number or fraction root gives exactly 0; each root with a square root in it is checked in exact form (the plain part and the square-root part both come to 0), and its two-decimal value is rounded from that; for each equation with no real roots the script checks that b² − 4ac is below zero and that the lowest value the left side takes, at x = −b/2a, is above zero. The names used (standard form, factoring, the quadratic formula, the discriminant b² − 4ac, a repeated or double root, no real roots when the discriminant is negative) follow Wikipedia, Quadratic equation, https://en.wikipedia.org/wiki/Quadratic_equation , read 2026-09-30. Licence: own work