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Solve -2x² + 12x - 12 = 0

The solutions of -2x² + 12x - 12 = 0 are x = 3 + √3 or x = 3 - √3, that is x ≈ 1.2679492 and x ≈ 4.7320508.

x² +x += 0
−2x2+12x−12=0⇒x=3+3, 3−3-2x^2 + 12x - 12 = 0 \Rightarrow x = 3 + \sqrt{3},\ 3 - \sqrt{3}
Discriminant D = 48 · x ≈ 1.26795, 4.73205

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

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Graph of y = -2x² + 12x - 12-11234567-30-25-20-15-10-5510x = 1.2679492x = 4.7320508vertexy = -2x² + 12x - 12
Graph of y = -2x² + 12x - 12. The curve crosses the x-axis at x = 1.2679492 and x = 4.7320508; the vertex is at (3, 6).

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 = -12.

    −2x2+12x−12=0-2x^2 + 12x - 12 = 0
  2. 2

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

    D=122−4⋅(−2)⋅(−12)=144−96=48D = 12^2 - 4\cdot(-2)\cdot(-12) = 144 - 96 = 48
  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±48−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{48}}{-4}
  5. 5

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

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

    As decimals: x ≈ 1.2679492 and x ≈ 4.7320508.

  7. 7

    Check numerically: for x ≈ 1.2679492 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 12x - 12 = 0\ \Rightarrow\ x = 3 + \sqrt{3},\ 3 - \sqrt{3} \]
\[ -2x^2 + 12x - 12 = 0 \]
\[ D = 12^2 - 4\cdot(-2)\cdot(-12) = 144 - 96 = 48 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{48}}{-4} \]
\[ x = 3 + \sqrt{3}\quad\text{or}\quad x = 3 - \sqrt{3} \]

The parabola y = -2x² + 12x - 12

xy
0-12
1-2
24
36
44
5-2
6-12

Questions

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

The solutions of -2x² + 12x - 12 = 0 are x = 3 + √3 or x = 3 - √3, that is x ≈ 1.2679492 and x ≈ 4.7320508.

Does -2x² + 12x - 12 = 0 factor?

Not over the integers, because the discriminant 48 is not a perfect square. Use the quadratic formula.

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

(3, 6). 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 = 6.

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