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Solve x² - 9x - 6 = 0

The solutions of x² - 9x - 6 = 0 are x = (9 + √105)/2 or x = (9 - √105)/2, that is x ≈ 9.6234754 and x ≈ -0.62347538.

x² +x += 0
x2−9x−6=0⇒x=9+1052, 9−1052x^2 - 9x - 6 = 0 \Rightarrow x = \frac{9 + \sqrt{105}}{2},\ \frac{9 - \sqrt{105}}{2}
Discriminant D = 105 · x ≈ 9.62348, -0.623475

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

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Graph of y = x² - 9x - 6-224681012-30-20-10102030x = 9.6234754x = -0.62347538vertexy = x² - 9x - 6
Graph of y = x² - 9x - 6. The curve crosses the x-axis at x = 9.6234754 and x = -0.62347538; the vertex is at (4.5, -26.25).

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 = -9, c = -6.

    x2−9x−6=0x^2 - 9x - 6 = 0
  2. 2

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

    D=(−9)2−4⋅1⋅(−6)=81+24=105D = (-9)^2 - 4\cdot1\cdot(-6) = 81 + 24 = 105
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=9±1052x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{105}}{2}
  5. 5

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

    x=9+1052orx=9−1052x = \frac{9 + \sqrt{105}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{105}}{2}
  6. 6

    As decimals: x ≈ 9.6234754 and x ≈ -0.62347538.

  7. 7

    Check numerically: for x ≈ 9.6234754 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 9x - 6 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{105}}{2},\ \frac{9 - \sqrt{105}}{2} \]
\[ x^2 - 9x - 6 = 0 \]
\[ D = (-9)^2 - 4\cdot1\cdot(-6) = 81 + 24 = 105 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{105}}{2} \]
\[ x = \frac{9 + \sqrt{105}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{105}}{2} \]

The parabola y = x² - 9x - 6

xy
1.5-17.25
2.5-22.25
3.5-25.25
4.5-26.25
5.5-25.25
6.5-22.25
7.5-17.25

Questions

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

The solutions of x² - 9x - 6 = 0 are x = (9 + √105)/2 or x = (9 - √105)/2, that is x ≈ 9.6234754 and x ≈ -0.62347538.

Does x² - 9x - 6 = 0 factor?

Not over the integers, because the discriminant 105 is not a perfect square. Use the quadratic formula.

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

(9/2, -105/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 9, product = c/a = -6.

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    Solve x² - 9x - 6 = 0: x = (9 + √105)/2 or x = (9 - √105)/2 (steps) | WordGPT