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Solve x² - 10x - 10 = 0

The solutions of x² - 10x - 10 = 0 are x = 5 + √35 or x = 5 - √35, that is x ≈ 10.91608 and x ≈ -0.91607978.

x² +x += 0
x2−10x−10=0⇒x=5+35, 5−35x^2 - 10x - 10 = 0 \Rightarrow x = 5 + \sqrt{35},\ 5 - \sqrt{35}
Discriminant D = 140 · x ≈ 10.9161, -0.91608

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

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Graph of y = x² - 10x - 10-224681012-40-30-20-1010203040x = 10.91608x = -0.91607978vertexy = x² - 10x - 10
Graph of y = x² - 10x - 10. The curve crosses the x-axis at x = 10.91608 and x = -0.91607978; the vertex is at (5, -35).

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 = -10.

    x2−10x−10=0x^2 - 10x - 10 = 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⋅(−10)=100+40=140D = (-10)^2 - 4\cdot1\cdot(-10) = 100 + 40 = 140
  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±1402x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{140}}{2}
  5. 5

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

    x=5+35orx=5−35x = 5 + \sqrt{35}\quad\text{or}\quad x = 5 - \sqrt{35}
  6. 6

    As decimals: x ≈ 10.91608 and x ≈ -0.91607978.

  7. 7

    Check numerically: for x ≈ 10.91608 the left side is -1.4210855e-14 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x - 10 = 0\ \Rightarrow\ x = 5 + \sqrt{35},\ 5 - \sqrt{35} \]
\[ x^2 - 10x - 10 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot(-10) = 100 + 40 = 140 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{140}}{2} \]
\[ x = 5 + \sqrt{35}\quad\text{or}\quad x = 5 - \sqrt{35} \]

The parabola y = x² - 10x - 10

xy
2-26
3-31
4-34
5-35
6-34
7-31
8-26

Questions

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

The solutions of x² - 10x - 10 = 0 are x = 5 + √35 or x = 5 - √35, that is x ≈ 10.91608 and x ≈ -0.91607978.

Does x² - 10x - 10 = 0 factor?

Not over the integers, because the discriminant 140 is not a perfect square. Use the quadratic formula.

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

(5, -35). 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 = -10.

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    Solve x² - 10x - 10 = 0: x = 5 + √35 or x = 5 - √35 (steps) | WordGPT