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Solve x² - 8x + 6 = 0

The solutions of x² - 8x + 6 = 0 are x = 4 + √10 or x = 4 - √10, that is x ≈ 7.1622777 and x ≈ 0.83772234.

x² +x += 0
x2−8x+6=0⇒x=4+10, 4−10x^2 - 8x + 6 = 0 \Rightarrow x = 4 + \sqrt{10},\ 4 - \sqrt{10}
Discriminant D = 40 · x ≈ 7.16228, 0.837722

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

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Graph of y = x² - 8x + 62468-10-55101520x = 7.1622777x = 0.83772234vertexy = x² - 8x + 6
Graph of y = x² - 8x + 6. The curve crosses the x-axis at x = 7.1622777 and x = 0.83772234; the vertex is at (4, -10).

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 = -8, c = 6.

    x2−8x+6=0x^2 - 8x + 6 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−8)2−4⋅1⋅6=64−24=40D = (-8)^2 - 4\cdot1\cdot6 = 64 - 24 = 40
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=8±402x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{40}}{2}
  5. 5

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

    x=4+10orx=4−10x = 4 + \sqrt{10}\quad\text{or}\quad x = 4 - \sqrt{10}
  6. 6

    As decimals: x ≈ 7.1622777 and x ≈ 0.83772234.

  7. 7

    Check numerically: for x ≈ 7.1622777 the left side is 7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 8x + 6 = 0\ \Rightarrow\ x = 4 + \sqrt{10},\ 4 - \sqrt{10} \]
\[ x^2 - 8x + 6 = 0 \]
\[ D = (-8)^2 - 4\cdot1\cdot6 = 64 - 24 = 40 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{40}}{2} \]
\[ x = 4 + \sqrt{10}\quad\text{or}\quad x = 4 - \sqrt{10} \]

The parabola y = x² - 8x + 6

xy
1-1
2-6
3-9
4-10
5-9
6-6
7-1

Questions

What are the solutions of x² - 8x + 6 = 0?

The solutions of x² - 8x + 6 = 0 are x = 4 + √10 or x = 4 - √10, that is x ≈ 7.1622777 and x ≈ 0.83772234.

Does x² - 8x + 6 = 0 factor?

Not over the integers, because the discriminant 40 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 8x + 6?

(4, -10). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = 6.

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