WordGPT

Solve x² - 9x - 2 = 0

The solutions of x² - 9x - 2 = 0 are x = (9 + √89)/2 or x = (9 - √89)/2, that is x ≈ 9.2169906 and x ≈ -0.21699057.

x² +x += 0
x2−9x−2=0⇒x=9+892, 9−892x^2 - 9x - 2 = 0 \Rightarrow x = \frac{9 + \sqrt{89}}{2},\ \frac{9 - \sqrt{89}}{2}
Discriminant D = 89 · x ≈ 9.21699, -0.216991

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

Explain it step by step in chatMake a short video of it
Graph of y = x² - 9x - 2-2246810-20-10102030x = 9.2169906x = -0.21699057vertexy = x² - 9x - 2
Graph of y = x² - 9x - 2. The curve crosses the x-axis at x = 9.2169906 and x = -0.21699057; the vertex is at (4.5, -22.25).

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 = -2.

    x2−9x−2=0x^2 - 9x - 2 = 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⋅(−2)=81+8=89D = (-9)^2 - 4\cdot1\cdot(-2) = 81 + 8 = 89
  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±892x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{89}}{2}
  5. 5

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

    x=9+892orx=9−892x = \frac{9 + \sqrt{89}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{89}}{2}
  6. 6

    As decimals: x ≈ 9.2169906 and x ≈ -0.21699057.

  7. 7

    Check numerically: for x ≈ 9.2169906 the left side is -1.4210855e-14 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 9x - 2 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{89}}{2},\ \frac{9 - \sqrt{89}}{2} \]
\[ x^2 - 9x - 2 = 0 \]
\[ D = (-9)^2 - 4\cdot1\cdot(-2) = 81 + 8 = 89 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{89}}{2} \]
\[ x = \frac{9 + \sqrt{89}}{2}\quad\text{or}\quad x = \frac{9 - \sqrt{89}}{2} \]

The parabola y = x² - 9x - 2

xy
1.5-13.25
2.5-18.25
3.5-21.25
4.5-22.25
5.5-21.25
6.5-18.25
7.5-13.25

Questions

What are the solutions of x² - 9x - 2 = 0?

The solutions of x² - 9x - 2 = 0 are x = (9 + √89)/2 or x = (9 - √89)/2, that is x ≈ 9.2169906 and x ≈ -0.21699057.

Does x² - 9x - 2 = 0 factor?

Not over the integers, because the discriminant 89 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 9x - 2?

(9/2, -89/4). 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 = -2.

Similar equations

    Solve x² - 9x - 2 = 0: x = (9 + √89)/2 or x = (9 - √89)/2 (steps) | WordGPT