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Solve x² - 10x - 7 = 0

The solutions of x² - 10x - 7 = 0 are x = 5 + 4√2 or x = 5 - 4√2, that is x ≈ 10.656854 and x ≈ -0.65685425.

x² +x += 0
x2−10x−7=0⇒x=5+42, 5−42x^2 - 10x - 7 = 0 \Rightarrow x = 5 + 4\sqrt{2},\ 5 - 4\sqrt{2}
Discriminant D = 128 · x ≈ 10.6569, -0.656854

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

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Graph of y = x² - 10x - 7-224681012-40-30-20-1010203040x = 10.656854x = -0.65685425vertexy = x² - 10x - 7
Graph of y = x² - 10x - 7. The curve crosses the x-axis at x = 10.656854 and x = -0.65685425; the vertex is at (5, -32).

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 = -7.

    x2−10x−7=0x^2 - 10x - 7 = 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⋅(−7)=100+28=128D = (-10)^2 - 4\cdot1\cdot(-7) = 100 + 28 = 128
  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±1282x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{128}}{2}
  5. 5

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

    x=5+42orx=5−42x = 5 + 4\sqrt{2}\quad\text{or}\quad x = 5 - 4\sqrt{2}
  6. 6

    As decimals: x ≈ 10.656854 and x ≈ -0.65685425.

  7. 7

    Check numerically: for x ≈ 10.656854 the left side is -1.4210855e-14 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x - 7 = 0\ \Rightarrow\ x = 5 + 4\sqrt{2},\ 5 - 4\sqrt{2} \]
\[ x^2 - 10x - 7 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot(-7) = 100 + 28 = 128 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{128}}{2} \]
\[ x = 5 + 4\sqrt{2}\quad\text{or}\quad x = 5 - 4\sqrt{2} \]

The parabola y = x² - 10x - 7

xy
2-23
3-28
4-31
5-32
6-31
7-28
8-23

Questions

What are the solutions of x² - 10x - 7 = 0?

The solutions of x² - 10x - 7 = 0 are x = 5 + 4√2 or x = 5 - 4√2, that is x ≈ 10.656854 and x ≈ -0.65685425.

Does x² - 10x - 7 = 0 factor?

Not over the integers, because the discriminant 128 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 10x - 7?

(5, -32). 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 = -7.

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    Solve x² - 10x - 7 = 0: x = 5 + 4√2 or x = 5 - 4√2 (steps) | WordGPT