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Solve -x² - 7x - 12 = 0

The solutions of -x² - 7x - 12 = 0 are x = -4 or x = -3.

x² +x += 0
−x2−7x−12=0⇒x=−4, −3-x^2 - 7x - 12 = 0 \Rightarrow x = -4,\ -3
Discriminant D = 1 · 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 - 7x - 12-6-5-4-3-2-1-8-6-4-2x = -4x = -3vertexy = -x² - 7x - 12
Graph of y = -x² - 7x - 12. The curve crosses the x-axis at x = -4 and x = -3; the vertex is at (-3.5, 0.25).

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 = -7, c = -12.

    −x2−7x−12=0-x^2 - 7x - 12 = 0
  2. 2

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

    D=(−7)2−4⋅(−1)⋅(−12)=49−48=1D = (-7)^2 - 4\cdot(-1)\cdot(-12) = 49 - 48 = 1
  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=7±1−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{1}}{-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) + (-7)·(-4) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 - 7x - 12 = 0\ \Rightarrow\ x = -4,\ -3 \]
\[ -x^2 - 7x - 12 = 0 \]
\[ D = (-7)^2 - 4\cdot(-1)\cdot(-12) = 49 - 48 = 1 \]
\[ -\left(x + 4\right)\left(x + 3\right) = 0 \]
\[ x + 4 = 0 \Rightarrow x = -4\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{1}}{-2} \]
\[ x = -4\quad\text{or}\quad x = -3 \]

The parabola y = -x² - 7x - 12

xy
-6.5-8.75
-5.5-3.75
-4.5-0.75
-3.50.25
-2.5-0.75
-1.5-3.75
-0.5-8.75

Questions

What are the solutions of -x² - 7x - 12 = 0?

The solutions of -x² - 7x - 12 = 0 are x = -4 or x = -3.

Does -x² - 7x - 12 = 0 factor?

Yes: -1(x+4)(x+3) = 0.

What is the vertex of y = -x² - 7x - 12?

(-7/2, 1/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -7, product = c/a = 12.

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