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Solve x² - x - 12 = 0

The solutions of x² - x - 12 = 0 are x = 4 or x = -3.

x² +x += 0
x2−x−12=0⇒x=4, −3x^2 - x - 12 = 0 \Rightarrow x = 4,\ -3
Discriminant D = 49 · x ≈ 4, -3

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = x² - x - 12-4-2246-15-10-5510152025x = 4x = -3vertexy = x² - x - 12
Graph of y = x² - x - 12. The curve crosses the x-axis at x = 4 and x = -3; the vertex is at (0.5, -12.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = -1, c = -12.

    x2−x−12=0x^2 - x - 12 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−1)2−4⋅1⋅(−12)=1+48=49D = (-1)^2 - 4\cdot1\cdot(-12) = 1 + 48 = 49
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x−4)(x+3)=0\left(x - 4\right)\left(x + 3\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x−4=0⇒x=4x+3=0⇒x=−3x - 4 = 0 \Rightarrow x = 4\qquad x + 3 = 0 \Rightarrow x = -3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±492x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{49}}{2}
  7. 7

    Simplify to get the solutions:

    x=4orx=−3x = 4\quad\text{or}\quad x = -3
  8. 8

    Check by putting x = 4 back in: a·x² + b·x + c = 1·(16) + (-1)·(4) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - x - 12 = 0\ \Rightarrow\ x = 4,\ -3 \]
\[ x^2 - x - 12 = 0 \]
\[ D = (-1)^2 - 4\cdot1\cdot(-12) = 1 + 48 = 49 \]
\[ \left(x - 4\right)\left(x + 3\right) = 0 \]
\[ x - 4 = 0 \Rightarrow x = 4\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{49}}{2} \]
\[ x = 4\quad\text{or}\quad x = -3 \]

The parabola y = x² - x - 12

xy
-2.5-3.25
-1.5-8.25
-0.5-11.25
0.5-12.25
1.5-11.25
2.5-8.25
3.5-3.25

Questions

What are the solutions of x² - x - 12 = 0?

The solutions of x² - x - 12 = 0 are x = 4 or x = -3.

Does x² - x - 12 = 0 factor?

Yes: (x-4)(x+3) = 0.

What is the vertex of y = x² - x - 12?

(1/2, -49/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -12.

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