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Solve x² - 10x + 10 = 0

The solutions of x² - 10x + 10 = 0 are x = 5 + √15 or x = 5 - √15, that is x ≈ 8.8729833 and x ≈ 1.1270167.

x² +x += 0
x2−10x+10=0⇒x=5+15, 5−15x^2 - 10x + 10 = 0 \Rightarrow x = 5 + \sqrt{15},\ 5 - \sqrt{15}
Discriminant D = 60 · x ≈ 8.87298, 1.12702

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

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Graph of y = x² - 10x + 10246810-20-101020x = 8.8729833x = 1.1270167vertexy = x² - 10x + 10
Graph of y = x² - 10x + 10. The curve crosses the x-axis at x = 8.8729833 and x = 1.1270167; the vertex is at (5, -15).

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=60D = (-10)^2 - 4\cdot1\cdot10 = 100 - 40 = 60
  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±602x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{60}}{2}
  5. 5

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

    x=5+15orx=5−15x = 5 + \sqrt{15}\quad\text{or}\quad x = 5 - \sqrt{15}
  6. 6

    As decimals: x ≈ 8.8729833 and x ≈ 1.1270167.

  7. 7

    Check numerically: for x ≈ 8.8729833 the left side is -1.4210855e-14 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x + 10 = 0\ \Rightarrow\ x = 5 + \sqrt{15},\ 5 - \sqrt{15} \]
\[ x^2 - 10x + 10 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot10 = 100 - 40 = 60 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{60}}{2} \]
\[ x = 5 + \sqrt{15}\quad\text{or}\quad x = 5 - \sqrt{15} \]

The parabola y = x² - 10x + 10

xy
2-6
3-11
4-14
5-15
6-14
7-11
8-6

Questions

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

The solutions of x² - 10x + 10 = 0 are x = 5 + √15 or x = 5 - √15, that is x ≈ 8.8729833 and x ≈ 1.1270167.

Does x² - 10x + 10 = 0 factor?

Not over the integers, because the discriminant 60 is not a perfect square. Use the quadratic formula.

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

(5, -15). 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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