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Solve -2x² + 8x + 10 = 0

The solutions of -2x² + 8x + 10 = 0 are x = -1 or x = 5.

x² +x += 0
−2x2+8x+10=0⇒x=−1, 5-2x^2 + 8x + 10 = 0 \Rightarrow x = -1,\ 5
Discriminant D = 144 · x ≈ -1, 5

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

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Graph of y = -2x² + 8x + 10-2246-40-30-20-101020x = -1x = 5vertexy = -2x² + 8x + 10
Graph of y = -2x² + 8x + 10. The curve crosses the x-axis at x = -1 and x = 5; the vertex is at (2, 18).

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 = 8, c = 10.

    −2x2+8x+10=0-2x^2 + 8x + 10 = 0
  2. 2

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

    D=82−4⋅(−2)⋅10=64+80=144D = 8^2 - 4\cdot(-2)\cdot10 = 64 + 80 = 144
  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+1)(x−5)=0-2\left(x + 1\right)\left(x - 5\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+1=0⇒x=−1x−5=0⇒x=5x + 1 = 0 \Rightarrow x = -1\qquad x - 5 = 0 \Rightarrow x = 5
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±144−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{144}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−1orx=5x = -1\quad\text{or}\quad x = 5
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -2·(1) + 8·(-1) + 10 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 8x + 10 = 0\ \Rightarrow\ x = -1,\ 5 \]
\[ -2x^2 + 8x + 10 = 0 \]
\[ D = 8^2 - 4\cdot(-2)\cdot10 = 64 + 80 = 144 \]
\[ -2\left(x + 1\right)\left(x - 5\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x - 5 = 0 \Rightarrow x = 5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{144}}{-4} \]
\[ x = -1\quad\text{or}\quad x = 5 \]

The parabola y = -2x² + 8x + 10

xy
-10
010
116
218
316
410
50

Questions

What are the solutions of -2x² + 8x + 10 = 0?

The solutions of -2x² + 8x + 10 = 0 are x = -1 or x = 5.

Does -2x² + 8x + 10 = 0 factor?

Yes: -2(x+1)(x-5) = 0.

What is the vertex of y = -2x² + 8x + 10?

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

What are the sum and product of the roots?

Sum = −b/a = 4, product = c/a = -5.

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