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Solve -x² + 12x + 9 = 0

The solutions of -x² + 12x + 9 = 0 are x = 6 + 3√5 or x = 6 - 3√5, that is x ≈ -0.70820393 and x ≈ 12.708204.

x² +x += 0
−x2+12x+9=0⇒x=6+35, 6−35-x^2 + 12x + 9 = 0 \Rightarrow x = 6 + 3\sqrt{5},\ 6 - 3\sqrt{5}
Discriminant D = 180 · x ≈ -0.708204, 12.7082

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

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Graph of y = -x² + 12x + 9-22468101214-40-202040x = -0.70820393x = 12.708204vertexy = -x² + 12x + 9
Graph of y = -x² + 12x + 9. The curve crosses the x-axis at x = -0.70820393 and x = 12.708204; the vertex is at (6, 45).

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 = 9.

    −x2+12x+9=0-x^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⋅(−1)⋅9=144+36=180D = 12^2 - 4\cdot(-1)\cdot9 = 144 + 36 = 180
  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±180−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{180}}{-2}
  5. 5

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

    x=6+35orx=6−35x = 6 + 3\sqrt{5}\quad\text{or}\quad x = 6 - 3\sqrt{5}
  6. 6

    As decimals: x ≈ -0.70820393 and x ≈ 12.708204.

  7. 7

    Check numerically: for x ≈ -0.70820393 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 12x + 9 = 0\ \Rightarrow\ x = 6 + 3\sqrt{5},\ 6 - 3\sqrt{5} \]
\[ -x^2 + 12x + 9 = 0 \]
\[ D = 12^2 - 4\cdot(-1)\cdot9 = 144 + 36 = 180 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{180}}{-2} \]
\[ x = 6 + 3\sqrt{5}\quad\text{or}\quad x = 6 - 3\sqrt{5} \]

The parabola y = -x² + 12x + 9

xy
336
441
544
645
744
841
936

Questions

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

The solutions of -x² + 12x + 9 = 0 are x = 6 + 3√5 or x = 6 - 3√5, that is x ≈ -0.70820393 and x ≈ 12.708204.

Does -x² + 12x + 9 = 0 factor?

Not over the integers, because the discriminant 180 is not a perfect square. Use the quadratic formula.

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

(6, 45). 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 = -9.

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    Solve -x² + 12x + 9 = 0: x = 6 + 3√5 or x = 6 - 3√5 (steps) | WordGPT