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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, −6−35x^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 - 9-14-12-10-8-6-4-22-40-202040x = 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=0x^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\cdot1\cdot(-9) = 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±1802x = \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
\[ x^2 + 12x - 9 = 0\ \Rightarrow\ x = -6 + 3\sqrt{5},\ -6 - 3\sqrt{5} \]
\[ x^2 + 12x - 9 = 0 \]
\[ D = 12^2 - 4\cdot1\cdot(-9) = 144 + 36 = 180 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{180}}{2} \]
\[ x = -6 + 3\sqrt{5}\quad\text{or}\quad x = -6 - 3\sqrt{5} \]

The parabola y = x² + 12x - 9

xy
-9-36
-8-41
-7-44
-6-45
-5-44
-4-41
-3-36

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 upwards, so this is the minimum.

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