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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, 1-x^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 + 12-14-12-10-8-6-4-22-40-202040x = -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=0-x^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\cdot(-1)\cdot12 = 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±169−2x = \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
\[ -x^2 - 11x + 12 = 0\ \Rightarrow\ x = -12,\ 1 \]
\[ -x^2 - 11x + 12 = 0 \]
\[ D = (-11)^2 - 4\cdot(-1)\cdot12 = 121 + 48 = 169 \]
\[ -\left(x + 12\right)\left(x - 1\right) = 0 \]
\[ x + 12 = 0 \Rightarrow x = -12\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{-2} \]
\[ x = -12\quad\text{or}\quad x = 1 \]

The parabola y = -x² - 11x + 12

xy
-8.533.25
-7.538.25
-6.541.25
-5.542.25
-4.541.25
-3.538.25
-2.533.25

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

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

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

What are the sum and product of the roots?

Sum = −b/a = -11, product = c/a = -12.

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