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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 ≈ 10.744563 and x ≈ -0.74456265.

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 ≈ 10.7446, -0.744563

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

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Graph of y = x² - 10x - 8-224681012-40-30-20-1010203040x = 10.744563x = -0.74456265vertexy = x² - 10x - 8
Graph of y = x² - 10x - 8. The curve crosses the x-axis at x = 10.744563 and x = -0.74456265; 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=(−10)2−4⋅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 ≈ 10.744563 and x ≈ -0.74456265.

  7. 7

    Check numerically: for x ≈ 10.744563 the left side is 0 (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
2-24
3-29
4-32
5-33
6-32
7-29
8-24

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 ≈ 10.744563 and x ≈ -0.74456265.

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