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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 ≈ 12.708204 and x ≈ -0.70820393.

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 ≈ 12.7082, -0.708204

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 = 12.708204x = -0.70820393vertexy = x² - 12x - 9
Graph of y = x² - 12x - 9. The curve crosses the x-axis at x = 12.708204 and x = -0.70820393; 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=(−12)2−4⋅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 ≈ 12.708204 and x ≈ -0.70820393.

  7. 7

    Check numerically: for x ≈ 12.708204 the left side is 2.8421709e-14 (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
3-36
4-41
5-44
6-45
7-44
8-41
9-36

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 ≈ 12.708204 and x ≈ -0.70820393.

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