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Solve 2x² - 9x - 9 = 0

The solutions of 2x² - 9x - 9 = 0 are x = (9 + 3√17)/4 or x = (9 - 3√17)/4, that is x ≈ 5.3423292 and x ≈ -0.84232922.

x² +x += 0
2x2−9x−9=0⇒x=9+3174, 9−31742x^2 - 9x - 9 = 0 \Rightarrow x = \frac{9 + 3\sqrt{17}}{4},\ \frac{9 - 3\sqrt{17}}{4}
Discriminant D = 153 · x ≈ 5.34233, -0.842329

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

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Graph of y = 2x² - 9x - 9-2246-20-1010203040x = 5.3423292x = -0.84232922vertexy = 2x² - 9x - 9
Graph of y = 2x² - 9x - 9. The curve crosses the x-axis at x = 5.3423292 and x = -0.84232922; the vertex is at (2.25, -19.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = -9, c = -9.

    2x2−9x−9=02x^2 - 9x - 9 = 0
  2. 2

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

    D=(−9)2−4⋅2⋅(−9)=81+72=153D = (-9)^2 - 4\cdot2\cdot(-9) = 81 + 72 = 153
  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±1534x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{153}}{4}
  5. 5

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

    x=9+3174orx=9−3174x = \frac{9 + 3\sqrt{17}}{4}\quad\text{or}\quad x = \frac{9 - 3\sqrt{17}}{4}
  6. 6

    As decimals: x ≈ 5.3423292 and x ≈ -0.84232922.

  7. 7

    Check numerically: for x ≈ 5.3423292 the left side is -7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 9x - 9 = 0\ \Rightarrow\ x = \frac{9 + 3\sqrt{17}}{4},\ \frac{9 - 3\sqrt{17}}{4} \]
\[ 2x^2 - 9x - 9 = 0 \]
\[ D = (-9)^2 - 4\cdot2\cdot(-9) = 81 + 72 = 153 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{153}}{4} \]
\[ x = \frac{9 + 3\sqrt{17}}{4}\quad\text{or}\quad x = \frac{9 - 3\sqrt{17}}{4} \]

The parabola y = 2x² - 9x - 9

xy
-0.75-1.125
0.25-11.125
1.25-17.125
2.25-19.125
3.25-17.125
4.25-11.125
5.25-1.125

Questions

What are the solutions of 2x² - 9x - 9 = 0?

The solutions of 2x² - 9x - 9 = 0 are x = (9 + 3√17)/4 or x = (9 - 3√17)/4, that is x ≈ 5.3423292 and x ≈ -0.84232922.

Does 2x² - 9x - 9 = 0 factor?

Not over the integers, because the discriminant 153 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 9x - 9?

(9/4, -153/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 9/2, product = c/a = -9/2.

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    Solve 2x² - 9x - 9 = 0: x = (9 + 3√17)/4 or x = (9 - 3√17)/4 (steps) | WordGPT