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Solve x² - 9x - 1 = 0

The solutions of x² - 9x - 1 = 0 are x = (9 + √85)/2 or x = (9 - √85)/2, that is x ≈ 9.1097722 and x ≈ -0.10977223.

x² +x += 0
x2−9x−1=0⇒x=9+852, 9−852x^2 - 9x - 1 = 0 \Rightarrow x = \frac{9 + \sqrt{85}}{2},\ \frac{9 - \sqrt{85}}{2}
Discriminant D = 85 · x ≈ 9.10977, -0.109772

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

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Graph of y = x² - 9x - 1-2246810-20-10102030x = 9.1097722x = -0.10977223vertexy = x² - 9x - 1
Graph of y = x² - 9x - 1. The curve crosses the x-axis at x = 9.1097722 and x = -0.10977223; the vertex is at (4.5, -21.25).

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 = -1.

    x2−9x−1=0x^2 - 9x - 1 = 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⋅(−1)=81+4=85D = (-9)^2 - 4\cdot1\cdot(-1) = 81 + 4 = 85
  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±852x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{85}}{2}
  5. 5

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

    x=9+852orx=9−852x = \frac{9 + \sqrt{85}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{85}}{2}
  6. 6

    As decimals: x ≈ 9.1097722 and x ≈ -0.10977223.

  7. 7

    Check numerically: for x ≈ 9.1097722 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 9x - 1 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{85}}{2},\ \frac{9 - \sqrt{85}}{2} \]
\[ x^2 - 9x - 1 = 0 \]
\[ D = (-9)^2 - 4\cdot1\cdot(-1) = 81 + 4 = 85 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{85}}{2} \]
\[ x = \frac{9 + \sqrt{85}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{85}}{2} \]

The parabola y = x² - 9x - 1

xy
1.5-12.25
2.5-17.25
3.5-20.25
4.5-21.25
5.5-20.25
6.5-17.25
7.5-12.25

Questions

What are the solutions of x² - 9x - 1 = 0?

The solutions of x² - 9x - 1 = 0 are x = (9 + √85)/2 or x = (9 - √85)/2, that is x ≈ 9.1097722 and x ≈ -0.10977223.

Does x² - 9x - 1 = 0 factor?

Not over the integers, because the discriminant 85 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 9x - 1?

(9/2, -85/4). 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 = -1.

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    Solve x² - 9x - 1 = 0: x = (9 + √85)/2 or x = (9 - √85)/2 (steps) | WordGPT