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Solve -2x² + 10x - 12 = 0

The solutions of -2x² + 10x - 12 = 0 are x = 2 or x = 3.

x² +x += 0
−2x2+10x−12=0⇒x=2, 3-2x^2 + 10x - 12 = 0 \Rightarrow x = 2,\ 3
Discriminant D = 4 · x ≈ 2, 3

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

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Graph of y = -2x² + 10x - 1212345-15-10-5x = 2x = 3vertexy = -2x² + 10x - 12
Graph of y = -2x² + 10x - 12. The curve crosses the x-axis at x = 2 and x = 3; the vertex is at (2.5, 0.5).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = 10, c = -12.

    −2x2+10x−12=0-2x^2 + 10x - 12 = 0
  2. 2

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

    D=102−4⋅(−2)⋅(−12)=100−96=4D = 10^2 - 4\cdot(-2)\cdot(-12) = 100 - 96 = 4
  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:

    −2(x−2)(x−3)=0-2\left(x - 2\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−2=0⇒x=2x−3=0⇒x=3x - 2 = 0 \Rightarrow x = 2\qquad x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−10±4−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{4}}{-4}
  7. 7

    Simplify to get the solutions:

    x=2orx=3x = 2\quad\text{or}\quad x = 3
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = -2·(4) + 10·(2) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 10x - 12 = 0\ \Rightarrow\ x = 2,\ 3 \]
\[ -2x^2 + 10x - 12 = 0 \]
\[ D = 10^2 - 4\cdot(-2)\cdot(-12) = 100 - 96 = 4 \]
\[ -2\left(x - 2\right)\left(x - 3\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{4}}{-4} \]
\[ x = 2\quad\text{or}\quad x = 3 \]

The parabola y = -2x² + 10x - 12

xy
-0.5-17.5
0.5-7.5
1.5-1.5
2.50.5
3.5-1.5
4.5-7.5
5.5-17.5

Questions

What are the solutions of -2x² + 10x - 12 = 0?

The solutions of -2x² + 10x - 12 = 0 are x = 2 or x = 3.

Does -2x² + 10x - 12 = 0 factor?

Yes: -2(x-2)(x-3) = 0.

What is the vertex of y = -2x² + 10x - 12?

(5/2, 1/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 5, product = c/a = 6.

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