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Solve x² - 10x + 6 = 0

The solutions of x² - 10x + 6 = 0 are x = 5 + √19 or x = 5 - √19, that is x ≈ 9.3588989 and x ≈ 0.64110106.

x² +x += 0
x2−10x+6=0⇒x=5+19, 5−19x^2 - 10x + 6 = 0 \Rightarrow x = 5 + \sqrt{19},\ 5 - \sqrt{19}
Discriminant D = 76 · x ≈ 9.3589, 0.641101

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

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Graph of y = x² - 10x + 6246810-20-10102030x = 9.3588989x = 0.64110106vertexy = x² - 10x + 6
Graph of y = x² - 10x + 6. The curve crosses the x-axis at x = 9.3588989 and x = 0.64110106; the vertex is at (5, -19).

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 = 6.

    x2−10x+6=0x^2 - 10x + 6 = 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⋅6=100−24=76D = (-10)^2 - 4\cdot1\cdot6 = 100 - 24 = 76
  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±762x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{76}}{2}
  5. 5

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

    x=5+19orx=5−19x = 5 + \sqrt{19}\quad\text{or}\quad x = 5 - \sqrt{19}
  6. 6

    As decimals: x ≈ 9.3588989 and x ≈ 0.64110106.

  7. 7

    Check numerically: for x ≈ 9.3588989 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x + 6 = 0\ \Rightarrow\ x = 5 + \sqrt{19},\ 5 - \sqrt{19} \]
\[ x^2 - 10x + 6 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot6 = 100 - 24 = 76 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{76}}{2} \]
\[ x = 5 + \sqrt{19}\quad\text{or}\quad x = 5 - \sqrt{19} \]

The parabola y = x² - 10x + 6

xy
2-10
3-15
4-18
5-19
6-18
7-15
8-10

Questions

What are the solutions of x² - 10x + 6 = 0?

The solutions of x² - 10x + 6 = 0 are x = 5 + √19 or x = 5 - √19, that is x ≈ 9.3588989 and x ≈ 0.64110106.

Does x² - 10x + 6 = 0 factor?

Not over the integers, because the discriminant 76 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 10x + 6?

(5, -19). 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 = 6.

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    Solve x² - 10x + 6 = 0: x = 5 + √19 or x = 5 - √19 (steps) | WordGPT