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Solve x² + 10x - 8 = 0

The solutions of x² + 10x - 8 = 0 are x = -5 + √33 or x = -5 - √33, that is x ≈ 0.74456265 and x ≈ -10.744563.

x² +x += 0
x2+10x−8=0⇒x=−5+33, −5−33x^2 + 10x - 8 = 0 \Rightarrow x = -5 + \sqrt{33},\ -5 - \sqrt{33}
Discriminant D = 132 · x ≈ 0.744563, -10.7446

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

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Graph of y = x² + 10x - 8-12-10-8-6-4-22-40-30-20-1010203040x = 0.74456265x = -10.744563vertexy = x² + 10x - 8
Graph of y = x² + 10x - 8. The curve crosses the x-axis at x = 0.74456265 and x = -10.744563; the vertex is at (-5, -33).

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 = 10, c = -8.

    x2+10x−8=0x^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⋅1⋅(−8)=100+32=132D = 10^2 - 4\cdot1\cdot(-8) = 100 + 32 = 132
  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±1322x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{132}}{2}
  5. 5

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

    x=−5+33orx=−5−33x = -5 + \sqrt{33}\quad\text{or}\quad x = -5 - \sqrt{33}
  6. 6

    As decimals: x ≈ 0.74456265 and x ≈ -10.744563.

  7. 7

    Check numerically: for x ≈ 0.74456265 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 10x - 8 = 0\ \Rightarrow\ x = -5 + \sqrt{33},\ -5 - \sqrt{33} \]
\[ x^2 + 10x - 8 = 0 \]
\[ D = 10^2 - 4\cdot1\cdot(-8) = 100 + 32 = 132 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{132}}{2} \]
\[ x = -5 + \sqrt{33}\quad\text{or}\quad x = -5 - \sqrt{33} \]

The parabola y = x² + 10x - 8

xy
-8-24
-7-29
-6-32
-5-33
-4-32
-3-29
-2-24

Questions

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

The solutions of x² + 10x - 8 = 0 are x = -5 + √33 or x = -5 - √33, that is x ≈ 0.74456265 and x ≈ -10.744563.

Does x² + 10x - 8 = 0 factor?

Not over the integers, because the discriminant 132 is not a perfect square. Use the quadratic formula.

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

(-5, -33). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -10, product = c/a = -8.

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    Solve x² + 10x - 8 = 0: x = -5 + √33 or x = -5 - √33 (steps) | WordGPT