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Solve x² - 10x + 3 = 0

The solutions of x² - 10x + 3 = 0 are x = 5 + √22 or x = 5 - √22, that is x ≈ 9.6904158 and x ≈ 0.30958424.

x² +x += 0
x2−10x+3=0⇒x=5+22, 5−22x^2 - 10x + 3 = 0 \Rightarrow x = 5 + \sqrt{22},\ 5 - \sqrt{22}
Discriminant D = 88 · x ≈ 9.69042, 0.309584

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

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Graph of y = x² - 10x + 3-224681012-20-10102030x = 9.6904158x = 0.30958424vertexy = x² - 10x + 3
Graph of y = x² - 10x + 3. The curve crosses the x-axis at x = 9.6904158 and x = 0.30958424; the vertex is at (5, -22).

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 = 3.

    x2−10x+3=0x^2 - 10x + 3 = 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⋅3=100−12=88D = (-10)^2 - 4\cdot1\cdot3 = 100 - 12 = 88
  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±882x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{88}}{2}
  5. 5

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

    x=5+22orx=5−22x = 5 + \sqrt{22}\quad\text{or}\quad x = 5 - \sqrt{22}
  6. 6

    As decimals: x ≈ 9.6904158 and x ≈ 0.30958424.

  7. 7

    Check numerically: for x ≈ 9.6904158 the left side is 1.4210855e-14 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x + 3 = 0\ \Rightarrow\ x = 5 + \sqrt{22},\ 5 - \sqrt{22} \]
\[ x^2 - 10x + 3 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot3 = 100 - 12 = 88 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{88}}{2} \]
\[ x = 5 + \sqrt{22}\quad\text{or}\quad x = 5 - \sqrt{22} \]

The parabola y = x² - 10x + 3

xy
2-13
3-18
4-21
5-22
6-21
7-18
8-13

Questions

What are the solutions of x² - 10x + 3 = 0?

The solutions of x² - 10x + 3 = 0 are x = 5 + √22 or x = 5 - √22, that is x ≈ 9.6904158 and x ≈ 0.30958424.

Does x² - 10x + 3 = 0 factor?

Not over the integers, because the discriminant 88 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 10x + 3?

(5, -22). 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 = 3.

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    Solve x² - 10x + 3 = 0: x = 5 + √22 or x = 5 - √22 (steps) | WordGPT