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Solve x² - 5x - 8 = 0

The solutions of x² - 5x - 8 = 0 are x = (5 + √57)/2 or x = (5 - √57)/2, that is x ≈ 6.2749172 and x ≈ -1.2749172.

x² +x += 0
x2−5x−8=0⇒x=5+572, 5−572x^2 - 5x - 8 = 0 \Rightarrow x = \frac{5 + \sqrt{57}}{2},\ \frac{5 - \sqrt{57}}{2}
Discriminant D = 57 · x ≈ 6.27492, -1.27492

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

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Graph of y = x² - 5x - 8-22468-101020x = 6.2749172x = -1.2749172vertexy = x² - 5x - 8
Graph of y = x² - 5x - 8. The curve crosses the x-axis at x = 6.2749172 and x = -1.2749172; the vertex is at (2.5, -14.25).

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 = -5, c = -8.

    x2−5x−8=0x^2 - 5x - 8 = 0
  2. 2

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

    D=(−5)2−4⋅1⋅(−8)=25+32=57D = (-5)^2 - 4\cdot1\cdot(-8) = 25 + 32 = 57
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=5±572x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{57}}{2}
  5. 5

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

    x=5+572orx=5−572x = \frac{5 + \sqrt{57}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{57}}{2}
  6. 6

    As decimals: x ≈ 6.2749172 and x ≈ -1.2749172.

  7. 7

    Check numerically: for x ≈ 6.2749172 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 5x - 8 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{57}}{2},\ \frac{5 - \sqrt{57}}{2} \]
\[ x^2 - 5x - 8 = 0 \]
\[ D = (-5)^2 - 4\cdot1\cdot(-8) = 25 + 32 = 57 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{57}}{2} \]
\[ x = \frac{5 + \sqrt{57}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{57}}{2} \]

The parabola y = x² - 5x - 8

xy
-0.5-5.25
0.5-10.25
1.5-13.25
2.5-14.25
3.5-13.25
4.5-10.25
5.5-5.25

Questions

What are the solutions of x² - 5x - 8 = 0?

The solutions of x² - 5x - 8 = 0 are x = (5 + √57)/2 or x = (5 - √57)/2, that is x ≈ 6.2749172 and x ≈ -1.2749172.

Does x² - 5x - 8 = 0 factor?

Not over the integers, because the discriminant 57 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 5x - 8?

(5/2, -57/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5, product = c/a = -8.

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    Solve x² - 5x - 8 = 0: x = (5 + √57)/2 or x = (5 - √57)/2 (steps) | WordGPT