WordGPT

Solve x² - 11x - 10 = 0

The solutions of x² - 11x - 10 = 0 are x = (11 + √161)/2 or x = (11 - √161)/2, that is x ≈ 11.844289 and x ≈ -0.84428877.

x² +x += 0
x2−11x−10=0⇒x=11+1612, 11−1612x^2 - 11x - 10 = 0 \Rightarrow x = \frac{11 + \sqrt{161}}{2},\ \frac{11 - \sqrt{161}}{2}
Discriminant D = 161 · x ≈ 11.8443, -0.844289

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 - 11x - 10-22468101214-40-202040x = 11.844289x = -0.84428877vertexy = x² - 11x - 10
Graph of y = x² - 11x - 10. The curve crosses the x-axis at x = 11.844289 and x = -0.84428877; the vertex is at (5.5, -40.25).

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 = -11, c = -10.

    x2−11x−10=0x^2 - 11x - 10 = 0
  2. 2

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

    D=(−11)2−4⋅1⋅(−10)=121+40=161D = (-11)^2 - 4\cdot1\cdot(-10) = 121 + 40 = 161
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=11±1612x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{161}}{2}
  5. 5

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

    x=11+1612orx=11−1612x = \frac{11 + \sqrt{161}}{2}\quad\text{or}\quad x = \frac{11 - \sqrt{161}}{2}
  6. 6

    As decimals: x ≈ 11.844289 and x ≈ -0.84428877.

  7. 7

    Check numerically: for x ≈ 11.844289 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 11x - 10 = 0\ \Rightarrow\ x = \frac{11 + \sqrt{161}}{2},\ \frac{11 - \sqrt{161}}{2} \]
\[ x^2 - 11x - 10 = 0 \]
\[ D = (-11)^2 - 4\cdot1\cdot(-10) = 121 + 40 = 161 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{161}}{2} \]
\[ x = \frac{11 + \sqrt{161}}{2}\quad\text{or}\quad x = \frac{11 - \sqrt{161}}{2} \]

The parabola y = x² - 11x - 10

xy
2.5-31.25
3.5-36.25
4.5-39.25
5.5-40.25
6.5-39.25
7.5-36.25
8.5-31.25

Questions

What are the solutions of x² - 11x - 10 = 0?

The solutions of x² - 11x - 10 = 0 are x = (11 + √161)/2 or x = (11 - √161)/2, that is x ≈ 11.844289 and x ≈ -0.84428877.

Does x² - 11x - 10 = 0 factor?

Not over the integers, because the discriminant 161 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 11x - 10?

(11/2, -161/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 11, product = c/a = -10.

Similar equations

    Solve x² - 11x - 10 = 0: x = (11 + √161)/2 or x = (11 - √161)/2 (steps) | WordGPT