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Solve -2x² + 12x + 6 = 0

The solutions of -2x² + 12x + 6 = 0 are x = 3 + 2√3 or x = 3 - 2√3, that is x ≈ -0.46410162 and x ≈ 6.4641016.

x² +x += 0
−2x2+12x+6=0⇒x=3+23, 3−23-2x^2 + 12x + 6 = 0 \Rightarrow x = 3 + 2\sqrt{3},\ 3 - 2\sqrt{3}
Discriminant D = 192 · x ≈ -0.464102, 6.4641

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

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Graph of y = -2x² + 12x + 6-22468-50-40-30-20-10102030x = -0.46410162x = 6.4641016vertexy = -2x² + 12x + 6
Graph of y = -2x² + 12x + 6. The curve crosses the x-axis at x = -0.46410162 and x = 6.4641016; the vertex is at (3, 24).

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 = 6.

    −2x2+12x+6=0-2x^2 + 12x + 6 = 0
  2. 2

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

    D=122−4⋅(−2)⋅6=144+48=192D = 12^2 - 4\cdot(-2)\cdot6 = 144 + 48 = 192
  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±192−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{192}}{-4}
  5. 5

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

    x=3+23orx=3−23x = 3 + 2\sqrt{3}\quad\text{or}\quad x = 3 - 2\sqrt{3}
  6. 6

    As decimals: x ≈ -0.46410162 and x ≈ 6.4641016.

  7. 7

    Check numerically: for x ≈ -0.46410162 the left side is 2.6645353e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 12x + 6 = 0\ \Rightarrow\ x = 3 + 2\sqrt{3},\ 3 - 2\sqrt{3} \]
\[ -2x^2 + 12x + 6 = 0 \]
\[ D = 12^2 - 4\cdot(-2)\cdot6 = 144 + 48 = 192 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{192}}{-4} \]
\[ x = 3 + 2\sqrt{3}\quad\text{or}\quad x = 3 - 2\sqrt{3} \]

The parabola y = -2x² + 12x + 6

xy
06
116
222
324
422
516
66

Questions

What are the solutions of -2x² + 12x + 6 = 0?

The solutions of -2x² + 12x + 6 = 0 are x = 3 + 2√3 or x = 3 - 2√3, that is x ≈ -0.46410162 and x ≈ 6.4641016.

Does -2x² + 12x + 6 = 0 factor?

Not over the integers, because the discriminant 192 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² + 12x + 6?

(3, 24). 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 = -3.

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