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Solve x² - 10x - 2 = 0

The solutions of x² - 10x - 2 = 0 are x = 5 + 3√3 or x = 5 - 3√3, that is x ≈ 10.196152 and x ≈ -0.19615242.

x² +x += 0
x2−10x−2=0⇒x=5+33, 5−33x^2 - 10x - 2 = 0 \Rightarrow x = 5 + 3\sqrt{3},\ 5 - 3\sqrt{3}
Discriminant D = 108 · x ≈ 10.1962, -0.196152

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

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Graph of y = x² - 10x - 2-224681012-30-20-10102030x = 10.196152x = -0.19615242vertexy = x² - 10x - 2
Graph of y = x² - 10x - 2. The curve crosses the x-axis at x = 10.196152 and x = -0.19615242; the vertex is at (5, -27).

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 = -2.

    x2−10x−2=0x^2 - 10x - 2 = 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⋅(−2)=100+8=108D = (-10)^2 - 4\cdot1\cdot(-2) = 100 + 8 = 108
  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±1082x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{108}}{2}
  5. 5

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

    x=5+33orx=5−33x = 5 + 3\sqrt{3}\quad\text{or}\quad x = 5 - 3\sqrt{3}
  6. 6

    As decimals: x ≈ 10.196152 and x ≈ -0.19615242.

  7. 7

    Check numerically: for x ≈ 10.196152 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x - 2 = 0\ \Rightarrow\ x = 5 + 3\sqrt{3},\ 5 - 3\sqrt{3} \]
\[ x^2 - 10x - 2 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot(-2) = 100 + 8 = 108 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{108}}{2} \]
\[ x = 5 + 3\sqrt{3}\quad\text{or}\quad x = 5 - 3\sqrt{3} \]

The parabola y = x² - 10x - 2

xy
2-18
3-23
4-26
5-27
6-26
7-23
8-18

Questions

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

The solutions of x² - 10x - 2 = 0 are x = 5 + 3√3 or x = 5 - 3√3, that is x ≈ 10.196152 and x ≈ -0.19615242.

Does x² - 10x - 2 = 0 factor?

Not over the integers, because the discriminant 108 is not a perfect square. Use the quadratic formula.

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

(5, -27). 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 = -2.

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