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Solve x² - 8x - 7 = 0

The solutions of x² - 8x - 7 = 0 are x = 4 + √23 or x = 4 - √23, that is x ≈ 8.7958315 and x ≈ -0.79583152.

x² +x += 0
x2−8x−7=0⇒x=4+23, 4−23x^2 - 8x - 7 = 0 \Rightarrow x = 4 + \sqrt{23},\ 4 - \sqrt{23}
Discriminant D = 92 · x ≈ 8.79583, -0.795832

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

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Graph of y = x² - 8x - 7-2246810-30-20-10102030x = 8.7958315x = -0.79583152vertexy = x² - 8x - 7
Graph of y = x² - 8x - 7. The curve crosses the x-axis at x = 8.7958315 and x = -0.79583152; the vertex is at (4, -23).

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 = -7.

    x2−8x−7=0x^2 - 8x - 7 = 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⋅(−7)=64+28=92D = (-8)^2 - 4\cdot1\cdot(-7) = 64 + 28 = 92
  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±922x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{92}}{2}
  5. 5

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

    x=4+23orx=4−23x = 4 + \sqrt{23}\quad\text{or}\quad x = 4 - \sqrt{23}
  6. 6

    As decimals: x ≈ 8.7958315 and x ≈ -0.79583152.

  7. 7

    Check numerically: for x ≈ 8.7958315 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 8x - 7 = 0\ \Rightarrow\ x = 4 + \sqrt{23},\ 4 - \sqrt{23} \]
\[ x^2 - 8x - 7 = 0 \]
\[ D = (-8)^2 - 4\cdot1\cdot(-7) = 64 + 28 = 92 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{92}}{2} \]
\[ x = 4 + \sqrt{23}\quad\text{or}\quad x = 4 - \sqrt{23} \]

The parabola y = x² - 8x - 7

xy
1-14
2-19
3-22
4-23
5-22
6-19
7-14

Questions

What are the solutions of x² - 8x - 7 = 0?

The solutions of x² - 8x - 7 = 0 are x = 4 + √23 or x = 4 - √23, that is x ≈ 8.7958315 and x ≈ -0.79583152.

Does x² - 8x - 7 = 0 factor?

Not over the integers, because the discriminant 92 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 8x - 7?

(4, -23). 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 = -7.

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    Solve x² - 8x - 7 = 0: x = 4 + √23 or x = 4 - √23 (steps) | WordGPT