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Solve -x² + 8x - 12 = 0

The solutions of -x² + 8x - 12 = 0 are x = 2 or x = 6.

x² +x += 0
−x2+8x−12=0⇒x=2, 6-x^2 + 8x - 12 = 0 \Rightarrow x = 2,\ 6
Discriminant D = 16 · x ≈ 2, 6

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

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Graph of y = -x² + 8x - 1212345678-15-10-55x = 2x = 6vertexy = -x² + 8x - 12
Graph of y = -x² + 8x - 12. The curve crosses the x-axis at x = 2 and x = 6; the vertex is at (4, 4).

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 = 8, c = -12.

    −x2+8x−12=0-x^2 + 8x - 12 = 0
  2. 2

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

    D=82−4⋅(−1)⋅(−12)=64−48=16D = 8^2 - 4\cdot(-1)\cdot(-12) = 64 - 48 = 16
  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−2)(x−6)=0-\left(x - 2\right)\left(x - 6\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−2=0⇒x=2x−6=0⇒x=6x - 2 = 0 \Rightarrow x = 2\qquad x - 6 = 0 \Rightarrow x = 6
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±16−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{16}}{-2}
  7. 7

    Simplify to get the solutions:

    x=2orx=6x = 2\quad\text{or}\quad x = 6
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = -1·(4) + 8·(2) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 8x - 12 = 0\ \Rightarrow\ x = 2,\ 6 \]
\[ -x^2 + 8x - 12 = 0 \]
\[ D = 8^2 - 4\cdot(-1)\cdot(-12) = 64 - 48 = 16 \]
\[ -\left(x - 2\right)\left(x - 6\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad x - 6 = 0 \Rightarrow x = 6 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{16}}{-2} \]
\[ x = 2\quad\text{or}\quad x = 6 \]

The parabola y = -x² + 8x - 12

xy
1-5
20
33
44
53
60
7-5

Questions

What are the solutions of -x² + 8x - 12 = 0?

The solutions of -x² + 8x - 12 = 0 are x = 2 or x = 6.

Does -x² + 8x - 12 = 0 factor?

Yes: -1(x-2)(x-6) = 0.

What is the vertex of y = -x² + 8x - 12?

(4, 4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = 12.

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