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Solve 2x² - 12x - 9 = 0

The solutions of 2x² - 12x - 9 = 0 are x = (6 + 3√6)/2 or x = (6 - 3√6)/2, that is x ≈ 6.6742346 and x ≈ -0.67423461.

x² +x += 0
2x2−12x−9=0⇒x=6+362, 6−3622x^2 - 12x - 9 = 0 \Rightarrow x = \frac{6 + 3\sqrt{6}}{2},\ \frac{6 - 3\sqrt{6}}{2}
Discriminant D = 216 · x ≈ 6.67423, -0.674235

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

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Graph of y = 2x² - 12x - 9-22468-202040x = 6.6742346x = -0.67423461vertexy = 2x² - 12x - 9
Graph of y = 2x² - 12x - 9. The curve crosses the x-axis at x = 6.6742346 and x = -0.67423461; the vertex is at (3, -27).

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 = -12, c = -9.

    2x2−12x−9=02x^2 - 12x - 9 = 0
  2. 2

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

    D=(−12)2−4⋅2⋅(−9)=144+72=216D = (-12)^2 - 4\cdot2\cdot(-9) = 144 + 72 = 216
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=12±2164x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{216}}{4}
  5. 5

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

    x=6+362orx=6−362x = \frac{6 + 3\sqrt{6}}{2}\quad\text{or}\quad x = \frac{6 - 3\sqrt{6}}{2}
  6. 6

    As decimals: x ≈ 6.6742346 and x ≈ -0.67423461.

  7. 7

    Check numerically: for x ≈ 6.6742346 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x - 9 = 0\ \Rightarrow\ x = \frac{6 + 3\sqrt{6}}{2},\ \frac{6 - 3\sqrt{6}}{2} \]
\[ 2x^2 - 12x - 9 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot(-9) = 144 + 72 = 216 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{216}}{4} \]
\[ x = \frac{6 + 3\sqrt{6}}{2}\quad\text{or}\quad x = \frac{6 - 3\sqrt{6}}{2} \]

The parabola y = 2x² - 12x - 9

xy
0-9
1-19
2-25
3-27
4-25
5-19
6-9

Questions

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

The solutions of 2x² - 12x - 9 = 0 are x = (6 + 3√6)/2 or x = (6 - 3√6)/2, that is x ≈ 6.6742346 and x ≈ -0.67423461.

Does 2x² - 12x - 9 = 0 factor?

Not over the integers, because the discriminant 216 is not a perfect square. Use the quadratic formula.

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

(3, -27). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = -9/2.

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    Solve 2x² - 12x - 9 = 0: x = (6 + 3√6)/2 or x = (6 - 3√6)/2 (steps) | WordGPT