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Solve 2x² + 10x - 8 = 0

The solutions of 2x² + 10x - 8 = 0 are x = (-5 + √41)/2 or x = (-5 - √41)/2, that is x ≈ 0.70156212 and x ≈ -5.7015621.

x² +x += 0
2x2+10x−8=0⇒x=−5+412, −5−4122x^2 + 10x - 8 = 0 \Rightarrow x = \frac{-5 + \sqrt{41}}{2},\ \frac{-5 - \sqrt{41}}{2}
Discriminant D = 164 · x ≈ 0.701562, -5.70156

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

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Graph of y = 2x² + 10x - 8-8-6-4-22-20-1010203040x = 0.70156212x = -5.7015621vertexy = 2x² + 10x - 8
Graph of y = 2x² + 10x - 8. The curve crosses the x-axis at x = 0.70156212 and x = -5.7015621; the vertex is at (-2.5, -20.5).

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 = -8.

    2x2+10x−8=02x^2 + 10x - 8 = 0
  2. 2

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

    D=102−4⋅2⋅(−8)=100+64=164D = 10^2 - 4\cdot2\cdot(-8) = 100 + 64 = 164
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−10±1644x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{164}}{4}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−5+412orx=−5−412x = \frac{-5 + \sqrt{41}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{41}}{2}
  6. 6

    As decimals: x ≈ 0.70156212 and x ≈ -5.7015621.

  7. 7

    Check numerically: for x ≈ 0.70156212 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 10x - 8 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{41}}{2},\ \frac{-5 - \sqrt{41}}{2} \]
\[ 2x^2 + 10x - 8 = 0 \]
\[ D = 10^2 - 4\cdot2\cdot(-8) = 100 + 64 = 164 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{164}}{4} \]
\[ x = \frac{-5 + \sqrt{41}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{41}}{2} \]

The parabola y = 2x² + 10x - 8

xy
-5.5-2.5
-4.5-12.5
-3.5-18.5
-2.5-20.5
-1.5-18.5
-0.5-12.5
0.5-2.5

Questions

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

The solutions of 2x² + 10x - 8 = 0 are x = (-5 + √41)/2 or x = (-5 - √41)/2, that is x ≈ 0.70156212 and x ≈ -5.7015621.

Does 2x² + 10x - 8 = 0 factor?

Not over the integers, because the discriminant 164 is not a perfect square. Use the quadratic formula.

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

(-5/2, -41/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = -4.

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    Solve 2x² + 10x - 8 = 0: x = (-5 + √41)/2 or x = (-5 - √41)/2 (steps) | WordGPT