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Solve -x² + 7x - 12 = 0

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

x² +x += 0
−x2+7x−12=0⇒x=3, 4-x^2 + 7x - 12 = 0 \Rightarrow x = 3,\ 4
Discriminant D = 1 · x ≈ 3, 4

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

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Graph of y = -x² + 7x - 12123456-8-6-4-2x = 3x = 4vertexy = -x² + 7x - 12
Graph of y = -x² + 7x - 12. The curve crosses the x-axis at x = 3 and x = 4; 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=72−4⋅(−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−3)(x−4)=0-\left(x - 3\right)\left(x - 4\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−3=0⇒x=3x−4=0⇒x=4x - 3 = 0 \Rightarrow x = 3\qquad x - 4 = 0 \Rightarrow x = 4
  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=3orx=4x = 3\quad\text{or}\quad x = 4
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = -1·(9) + 7·(3) + (-12) = 0 ✓

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

The parabola y = -x² + 7x - 12

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

Questions

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

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

Does -x² + 7x - 12 = 0 factor?

Yes: -1(x-3)(x-4) = 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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