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Solve x² + 9x - 8 = 0

The solutions of x² + 9x - 8 = 0 are x = (-9 + √113)/2 or x = (-9 - √113)/2, that is x ≈ 0.81507291 and x ≈ -9.8150729.

x² +x += 0
x2+9x−8=0⇒x=−9+1132, −9−1132x^2 + 9x - 8 = 0 \Rightarrow x = \frac{-9 + \sqrt{113}}{2},\ \frac{-9 - \sqrt{113}}{2}
Discriminant D = 113 · x ≈ 0.815073, -9.81507

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

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Graph of y = x² + 9x - 8-12-10-8-6-4-22-30-20-10102030x = 0.81507291x = -9.8150729vertexy = x² + 9x - 8
Graph of y = x² + 9x - 8. The curve crosses the x-axis at x = 0.81507291 and x = -9.8150729; the vertex is at (-4.5, -28.25).

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 = 9, c = -8.

    x2+9x−8=0x^2 + 9x - 8 = 0
  2. 2

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

    D=92−4⋅1⋅(−8)=81+32=113D = 9^2 - 4\cdot1\cdot(-8) = 81 + 32 = 113
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−9±1132x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{113}}{2}
  5. 5

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

    x=−9+1132orx=−9−1132x = \frac{-9 + \sqrt{113}}{2}\quad\text{or}\quad x = \frac{-9 - \sqrt{113}}{2}
  6. 6

    As decimals: x ≈ 0.81507291 and x ≈ -9.8150729.

  7. 7

    Check numerically: for x ≈ 0.81507291 the left side is 3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 9x - 8 = 0\ \Rightarrow\ x = \frac{-9 + \sqrt{113}}{2},\ \frac{-9 - \sqrt{113}}{2} \]
\[ x^2 + 9x - 8 = 0 \]
\[ D = 9^2 - 4\cdot1\cdot(-8) = 81 + 32 = 113 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{113}}{2} \]
\[ x = \frac{-9 + \sqrt{113}}{2}\quad\text{or}\quad x = \frac{-9 - \sqrt{113}}{2} \]

The parabola y = x² + 9x - 8

xy
-7.5-19.25
-6.5-24.25
-5.5-27.25
-4.5-28.25
-3.5-27.25
-2.5-24.25
-1.5-19.25

Questions

What are the solutions of x² + 9x - 8 = 0?

The solutions of x² + 9x - 8 = 0 are x = (-9 + √113)/2 or x = (-9 - √113)/2, that is x ≈ 0.81507291 and x ≈ -9.8150729.

Does x² + 9x - 8 = 0 factor?

Not over the integers, because the discriminant 113 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 9x - 8?

(-9/2, -113/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -9, product = c/a = -8.

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    Solve x² + 9x - 8 = 0: x = (-9 + √113)/2 or x = (-9 - √113)/2 (steps) | WordGPT