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Solve 2x² - 9x - 12 = 0

The solutions of 2x² - 9x - 12 = 0 are x = (9 + √177)/4 or x = (9 - √177)/4, that is x ≈ 5.5760337 and x ≈ -1.0760337.

x² +x += 0
2x2−9x−12=0⇒x=9+1774, 9−17742x^2 - 9x - 12 = 0 \Rightarrow x = \frac{9 + \sqrt{177}}{4},\ \frac{9 - \sqrt{177}}{4}
Discriminant D = 177 · x ≈ 5.57603, -1.07603

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

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Graph of y = 2x² - 9x - 12-22468-30-20-101020304050x = 5.5760337x = -1.0760337vertexy = 2x² - 9x - 12
Graph of y = 2x² - 9x - 12. The curve crosses the x-axis at x = 5.5760337 and x = -1.0760337; the vertex is at (2.25, -22.125).

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 = -12.

    2x2−9x−12=02x^2 - 9x - 12 = 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⋅(−12)=81+96=177D = (-9)^2 - 4\cdot2\cdot(-12) = 81 + 96 = 177
  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±1774x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{177}}{4}
  5. 5

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

    x=9+1774orx=9−1774x = \frac{9 + \sqrt{177}}{4}\quad\text{or}\quad x = \frac{9 - \sqrt{177}}{4}
  6. 6

    As decimals: x ≈ 5.5760337 and x ≈ -1.0760337.

  7. 7

    Check numerically: for x ≈ 5.5760337 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 9x - 12 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{177}}{4},\ \frac{9 - \sqrt{177}}{4} \]
\[ 2x^2 - 9x - 12 = 0 \]
\[ D = (-9)^2 - 4\cdot2\cdot(-12) = 81 + 96 = 177 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{177}}{4} \]
\[ x = \frac{9 + \sqrt{177}}{4}\quad\text{or}\quad x = \frac{9 - \sqrt{177}}{4} \]

The parabola y = 2x² - 9x - 12

xy
-0.75-4.125
0.25-14.125
1.25-20.125
2.25-22.125
3.25-20.125
4.25-14.125
5.25-4.125

Questions

What are the solutions of 2x² - 9x - 12 = 0?

The solutions of 2x² - 9x - 12 = 0 are x = (9 + √177)/4 or x = (9 - √177)/4, that is x ≈ 5.5760337 and x ≈ -1.0760337.

Does 2x² - 9x - 12 = 0 factor?

Not over the integers, because the discriminant 177 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 9x - 12?

(9/4, -177/8). 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 = -6.

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    Solve 2x² - 9x - 12 = 0: x = (9 + √177)/4 or x = (9 - √177)/4 (steps) | WordGPT