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Solve -x² + 6x + 12 = 0

The solutions of -x² + 6x + 12 = 0 are x = 3 + √21 or x = 3 - √21, that is x ≈ -1.5825757 and x ≈ 7.5825757.

x² +x += 0
−x2+6x+12=0⇒x=3+21, 3−21-x^2 + 6x + 12 = 0 \Rightarrow x = 3 + \sqrt{21},\ 3 - \sqrt{21}
Discriminant D = 84 · x ≈ -1.58258, 7.58258

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

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Graph of y = -x² + 6x + 12-4-2246810-30-20-101020x = -1.5825757x = 7.5825757vertexy = -x² + 6x + 12
Graph of y = -x² + 6x + 12. The curve crosses the x-axis at x = -1.5825757 and x = 7.5825757; the vertex is at (3, 21).

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 = 6, c = 12.

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

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

    D=62−4⋅(−1)⋅12=36+48=84D = 6^2 - 4\cdot(-1)\cdot12 = 36 + 48 = 84
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−6±84−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{84}}{-2}
  5. 5

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

    x=3+21orx=3−21x = 3 + \sqrt{21}\quad\text{or}\quad x = 3 - \sqrt{21}
  6. 6

    As decimals: x ≈ -1.5825757 and x ≈ 7.5825757.

  7. 7

    Check numerically: for x ≈ -1.5825757 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 6x + 12 = 0\ \Rightarrow\ x = 3 + \sqrt{21},\ 3 - \sqrt{21} \]
\[ -x^2 + 6x + 12 = 0 \]
\[ D = 6^2 - 4\cdot(-1)\cdot12 = 36 + 48 = 84 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{84}}{-2} \]
\[ x = 3 + \sqrt{21}\quad\text{or}\quad x = 3 - \sqrt{21} \]

The parabola y = -x² + 6x + 12

xy
012
117
220
321
420
517
612

Questions

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

The solutions of -x² + 6x + 12 = 0 are x = 3 + √21 or x = 3 - √21, that is x ≈ -1.5825757 and x ≈ 7.5825757.

Does -x² + 6x + 12 = 0 factor?

Not over the integers, because the discriminant 84 is not a perfect square. Use the quadratic formula.

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

(3, 21). 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 = -12.

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    Solve -x² + 6x + 12 = 0: x = 3 + √21 or x = 3 - √21 (steps) | WordGPT