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Solve 2x² - 8x - 10 = 0

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

x² +x += 0
2x2−8x−10=0⇒x=5, −12x^2 - 8x - 10 = 0 \Rightarrow x = 5,\ -1
Discriminant D = 144 · x ≈ 5, -1

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

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Graph of y = 2x² - 8x - 10-2246-20-1010203040x = 5x = -1vertexy = 2x² - 8x - 10
Graph of y = 2x² - 8x - 10. The curve crosses the x-axis at x = 5 and x = -1; 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=02x^2 - 8x - 10 = 0
  2. 2

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

    D=(−8)2−4⋅2⋅(−10)=64+80=144D = (-8)^2 - 4\cdot2\cdot(-10) = 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−5)(x+1)=02\left(x - 5\right)\left(x + 1\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−5=0⇒x=5x+1=0⇒x=−1x - 5 = 0 \Rightarrow x = 5\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=8±1444x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{144}}{4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 5 back in: a·x² + b·x + c = 2·(25) + (-8)·(5) + (-10) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 8x - 10 = 0\ \Rightarrow\ x = 5,\ -1 \]
\[ 2x^2 - 8x - 10 = 0 \]
\[ D = (-8)^2 - 4\cdot2\cdot(-10) = 64 + 80 = 144 \]
\[ 2\left(x - 5\right)\left(x + 1\right) = 0 \]
\[ x - 5 = 0 \Rightarrow x = 5\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{144}}{4} \]
\[ x = 5\quad\text{or}\quad x = -1 \]

The parabola y = 2x² - 8x - 10

xy
-10
0-10
1-16
2-18
3-16
4-10
50

Questions

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

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

Does 2x² - 8x - 10 = 0 factor?

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

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

(2, -18). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 4, product = c/a = -5.

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