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Solve -x² - 12x + 12 = 0

The solutions of -x² - 12x + 12 = 0 are x = -6 + 4√3 or x = -6 - 4√3, that is x ≈ -12.928203 and x ≈ 0.92820323.

x² +x += 0
−x2−12x+12=0⇒x=−6+43, −6−43-x^2 - 12x + 12 = 0 \Rightarrow x = -6 + 4\sqrt{3},\ -6 - 4\sqrt{3}
Discriminant D = 192 · x ≈ -12.9282, 0.928203

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

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Graph of y = -x² - 12x + 12-14-12-10-8-6-4-22-40-202040x = -12.928203x = 0.92820323vertexy = -x² - 12x + 12
Graph of y = -x² - 12x + 12. The curve crosses the x-axis at x = -12.928203 and x = 0.92820323; the vertex is at (-6, 48).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = -12, c = 12.

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

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

    D=(−12)2−4⋅(−1)⋅12=144+48=192D = (-12)^2 - 4\cdot(-1)\cdot12 = 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±192−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{192}}{-2}
  5. 5

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

    x=−6+43orx=−6−43x = -6 + 4\sqrt{3}\quad\text{or}\quad x = -6 - 4\sqrt{3}
  6. 6

    As decimals: x ≈ -12.928203 and x ≈ 0.92820323.

  7. 7

    Check numerically: for x ≈ -12.928203 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 - 12x + 12 = 0\ \Rightarrow\ x = -6 + 4\sqrt{3},\ -6 - 4\sqrt{3} \]
\[ -x^2 - 12x + 12 = 0 \]
\[ D = (-12)^2 - 4\cdot(-1)\cdot12 = 144 + 48 = 192 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{192}}{-2} \]
\[ x = -6 + 4\sqrt{3}\quad\text{or}\quad x = -6 - 4\sqrt{3} \]

The parabola y = -x² - 12x + 12

xy
-939
-844
-747
-648
-547
-444
-339

Questions

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

The solutions of -x² - 12x + 12 = 0 are x = -6 + 4√3 or x = -6 - 4√3, that is x ≈ -12.928203 and x ≈ 0.92820323.

Does -x² - 12x + 12 = 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 = -x² - 12x + 12?

(-6, 48). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -12, product = c/a = -12.

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    Solve -x² - 12x + 12 = 0: x = -6 + 4√3 or x = -6 - 4√3 (steps) | WordGPT