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Solve x² - 6x - 9 = 0

The solutions of x² - 6x - 9 = 0 are x = 3 + 3√2 or x = 3 - 3√2, that is x ≈ 7.2426407 and x ≈ -1.2426407.

x² +x += 0
x2−6x−9=0⇒x=3+32, 3−32x^2 - 6x - 9 = 0 \Rightarrow x = 3 + 3\sqrt{2},\ 3 - 3\sqrt{2}
Discriminant D = 72 · x ≈ 7.24264, -1.24264

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

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Graph of y = x² - 6x - 9-22468-20-10102030x = 7.2426407x = -1.2426407vertexy = x² - 6x - 9
Graph of y = x² - 6x - 9. The curve crosses the x-axis at x = 7.2426407 and x = -1.2426407; the vertex is at (3, -18).

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 = -9.

    x2−6x−9=0x^2 - 6x - 9 = 0
  2. 2

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

    D=(−6)2−4⋅1⋅(−9)=36+36=72D = (-6)^2 - 4\cdot1\cdot(-9) = 36 + 36 = 72
  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±722x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{72}}{2}
  5. 5

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

    x=3+32orx=3−32x = 3 + 3\sqrt{2}\quad\text{or}\quad x = 3 - 3\sqrt{2}
  6. 6

    As decimals: x ≈ 7.2426407 and x ≈ -1.2426407.

  7. 7

    Check numerically: for x ≈ 7.2426407 the left side is -7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 6x - 9 = 0\ \Rightarrow\ x = 3 + 3\sqrt{2},\ 3 - 3\sqrt{2} \]
\[ x^2 - 6x - 9 = 0 \]
\[ D = (-6)^2 - 4\cdot1\cdot(-9) = 36 + 36 = 72 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{72}}{2} \]
\[ x = 3 + 3\sqrt{2}\quad\text{or}\quad x = 3 - 3\sqrt{2} \]

The parabola y = x² - 6x - 9

xy
0-9
1-14
2-17
3-18
4-17
5-14
6-9

Questions

What are the solutions of x² - 6x - 9 = 0?

The solutions of x² - 6x - 9 = 0 are x = 3 + 3√2 or x = 3 - 3√2, that is x ≈ 7.2426407 and x ≈ -1.2426407.

Does x² - 6x - 9 = 0 factor?

Not over the integers, because the discriminant 72 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 6x - 9?

(3, -18). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = -9.

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    Solve x² - 6x - 9 = 0: x = 3 + 3√2 or x = 3 - 3√2 (steps) | WordGPT