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Solve x² - 10x + 1 = 0

The solutions of x² - 10x + 1 = 0 are x = 5 + 2√6 or x = 5 - 2√6, that is x ≈ 9.8989795 and x ≈ 0.10102051.

x² +x += 0
x2−10x+1=0⇒x=5+26, 5−26x^2 - 10x + 1 = 0 \Rightarrow x = 5 + 2\sqrt{6},\ 5 - 2\sqrt{6}
Discriminant D = 96 · x ≈ 9.89898, 0.101021

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

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Graph of y = x² - 10x + 1-224681012-30-20-10102030x = 9.8989795x = 0.10102051vertexy = x² - 10x + 1
Graph of y = x² - 10x + 1. The curve crosses the x-axis at x = 9.8989795 and x = 0.10102051; the vertex is at (5, -24).

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 = 1.

    x2−10x+1=0x^2 - 10x + 1 = 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⋅1=100−4=96D = (-10)^2 - 4\cdot1\cdot1 = 100 - 4 = 96
  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±962x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{96}}{2}
  5. 5

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

    x=5+26orx=5−26x = 5 + 2\sqrt{6}\quad\text{or}\quad x = 5 - 2\sqrt{6}
  6. 6

    As decimals: x ≈ 9.8989795 and x ≈ 0.10102051.

  7. 7

    Check numerically: for x ≈ 9.8989795 the left side is -1.4210855e-14 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x + 1 = 0\ \Rightarrow\ x = 5 + 2\sqrt{6},\ 5 - 2\sqrt{6} \]
\[ x^2 - 10x + 1 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot1 = 100 - 4 = 96 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{96}}{2} \]
\[ x = 5 + 2\sqrt{6}\quad\text{or}\quad x = 5 - 2\sqrt{6} \]

The parabola y = x² - 10x + 1

xy
2-15
3-20
4-23
5-24
6-23
7-20
8-15

Questions

What are the solutions of x² - 10x + 1 = 0?

The solutions of x² - 10x + 1 = 0 are x = 5 + 2√6 or x = 5 - 2√6, that is x ≈ 9.8989795 and x ≈ 0.10102051.

Does x² - 10x + 1 = 0 factor?

Not over the integers, because the discriminant 96 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 10x + 1?

(5, -24). 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 = 1.

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    Solve x² - 10x + 1 = 0: x = 5 + 2√6 or x = 5 - 2√6 (steps) | WordGPT