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Solve -2x² + 12x - 9 = 0

The solutions of -2x² + 12x - 9 = 0 are x = (6 + 3√2)/2 or x = (6 - 3√2)/2, that is x ≈ 0.87867966 and x ≈ 5.1213203.

x² +x += 0
−2x2+12x−9=0⇒x=6+322, 6−322-2x^2 + 12x - 9 = 0 \Rightarrow x = \frac{6 + 3\sqrt{2}}{2},\ \frac{6 - 3\sqrt{2}}{2}
Discriminant D = 72 · x ≈ 0.87868, 5.12132

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

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Graph of y = -2x² + 12x - 9-11234567-30-20-1010x = 0.87867966x = 5.1213203vertexy = -2x² + 12x - 9
Graph of y = -2x² + 12x - 9. The curve crosses the x-axis at x = 0.87867966 and x = 5.1213203; the vertex is at (3, 9).

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=0-2x^2 + 12x - 9 = 0
  2. 2

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

    D=122−4⋅(−2)⋅(−9)=144−72=72D = 12^2 - 4\cdot(-2)\cdot(-9) = 144 - 72 = 72
  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±72−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{72}}{-4}
  5. 5

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

    x=6+322orx=6−322x = \frac{6 + 3\sqrt{2}}{2}\quad\text{or}\quad x = \frac{6 - 3\sqrt{2}}{2}
  6. 6

    As decimals: x ≈ 0.87867966 and x ≈ 5.1213203.

  7. 7

    Check numerically: for x ≈ 0.87867966 the left side is 1.7763568e-15 (zero up to rounding) ✓

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

The parabola y = -2x² + 12x - 9

xy
0-9
11
27
39
47
51
6-9

Questions

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

The solutions of -2x² + 12x - 9 = 0 are x = (6 + 3√2)/2 or x = (6 - 3√2)/2, that is x ≈ 0.87867966 and x ≈ 5.1213203.

Does -2x² + 12x - 9 = 0 factor?

Not over the integers, because the discriminant 72 is not a perfect square. Use the quadratic formula.

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

(3, 9). The parabola opens downwards, so this is the maximum.

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√2)/2 or x = (6 - 3√2)/2 (steps) | WordGPT