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Solve x² - 3x - 9 = 0

The solutions of x² - 3x - 9 = 0 are x = (3 + 3√5)/2 or x = (3 - 3√5)/2, that is x ≈ 4.854102 and x ≈ -1.854102.

x² +x += 0
x2−3x−9=0⇒x=3+352, 3−352x^2 - 3x - 9 = 0 \Rightarrow x = \frac{3 + 3\sqrt{5}}{2},\ \frac{3 - 3\sqrt{5}}{2}
Discriminant D = 45 · x ≈ 4.8541, -1.8541

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

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Graph of y = x² - 3x - 9-4-2246-15-10-5510152025x = 4.854102x = -1.854102vertexy = x² - 3x - 9
Graph of y = x² - 3x - 9. The curve crosses the x-axis at x = 4.854102 and x = -1.854102; the vertex is at (1.5, -11.25).

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 = -3, c = -9.

    x2−3x−9=0x^2 - 3x - 9 = 0
  2. 2

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

    D=(−3)2−4⋅1⋅(−9)=9+36=45D = (-3)^2 - 4\cdot1\cdot(-9) = 9 + 36 = 45
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=3±452x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{45}}{2}
  5. 5

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

    x=3+352orx=3−352x = \frac{3 + 3\sqrt{5}}{2}\quad\text{or}\quad x = \frac{3 - 3\sqrt{5}}{2}
  6. 6

    As decimals: x ≈ 4.854102 and x ≈ -1.854102.

  7. 7

    Check numerically: for x ≈ 4.854102 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 3x - 9 = 0\ \Rightarrow\ x = \frac{3 + 3\sqrt{5}}{2},\ \frac{3 - 3\sqrt{5}}{2} \]
\[ x^2 - 3x - 9 = 0 \]
\[ D = (-3)^2 - 4\cdot1\cdot(-9) = 9 + 36 = 45 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{45}}{2} \]
\[ x = \frac{3 + 3\sqrt{5}}{2}\quad\text{or}\quad x = \frac{3 - 3\sqrt{5}}{2} \]

The parabola y = x² - 3x - 9

xy
-1.5-2.25
-0.5-7.25
0.5-10.25
1.5-11.25
2.5-10.25
3.5-7.25
4.5-2.25

Questions

What are the solutions of x² - 3x - 9 = 0?

The solutions of x² - 3x - 9 = 0 are x = (3 + 3√5)/2 or x = (3 - 3√5)/2, that is x ≈ 4.854102 and x ≈ -1.854102.

Does x² - 3x - 9 = 0 factor?

Not over the integers, because the discriminant 45 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 3x - 9?

(3/2, -45/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = -9.

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