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Solve x² - 11x - 12 = 0

The solutions of x² - 11x - 12 = 0 are x = 12 or x = -1.

x² +x += 0
x2−11x−12=0⇒x=12, −1x^2 - 11x - 12 = 0 \Rightarrow x = 12,\ -1
Discriminant D = 169 · x ≈ 12, -1

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

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Graph of y = x² - 11x - 12-22468101214-40-202040x = 12x = -1vertexy = x² - 11x - 12
Graph of y = x² - 11x - 12. The curve crosses the x-axis at x = 12 and x = -1; the vertex is at (5.5, -42.25).

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 = -11, c = -12.

    x2−11x−12=0x^2 - 11x - 12 = 0
  2. 2

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

    D=(−11)2−4⋅1⋅(−12)=121+48=169D = (-11)^2 - 4\cdot1\cdot(-12) = 121 + 48 = 169
  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−12)(x+1)=0\left(x - 12\right)\left(x + 1\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−12=0⇒x=12x+1=0⇒x=−1x - 12 = 0 \Rightarrow x = 12\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±1692x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{2}
  7. 7

    Simplify to get the solutions:

    x=12orx=−1x = 12\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 12 back in: a·x² + b·x + c = 1·(144) + (-11)·(12) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - 11x - 12 = 0\ \Rightarrow\ x = 12,\ -1 \]
\[ x^2 - 11x - 12 = 0 \]
\[ D = (-11)^2 - 4\cdot1\cdot(-12) = 121 + 48 = 169 \]
\[ \left(x - 12\right)\left(x + 1\right) = 0 \]
\[ x - 12 = 0 \Rightarrow x = 12\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{2} \]
\[ x = 12\quad\text{or}\quad x = -1 \]

The parabola y = x² - 11x - 12

xy
2.5-33.25
3.5-38.25
4.5-41.25
5.5-42.25
6.5-41.25
7.5-38.25
8.5-33.25

Questions

What are the solutions of x² - 11x - 12 = 0?

The solutions of x² - 11x - 12 = 0 are x = 12 or x = -1.

Does x² - 11x - 12 = 0 factor?

Yes: (x-12)(x+1) = 0.

What is the vertex of y = x² - 11x - 12?

(11/2, -169/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 11, product = c/a = -12.

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