WordGPT

Solve -x² + 11x - 10 = 0

The solutions of -x² + 11x - 10 = 0 are x = 1 or x = 10.

x² +x += 0
−x2+11x−10=0⇒x=1, 10-x^2 + 11x - 10 = 0 \Rightarrow x = 1,\ 10
Discriminant D = 81 · x ≈ 1, 10

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 - 1024681012-30-20-101020x = 1x = 10vertexy = -x² + 11x - 10
Graph of y = -x² + 11x - 10. The curve crosses the x-axis at x = 1 and x = 10; the vertex is at (5.5, 20.25).

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=0-x^2 + 11x - 10 = 0
  2. 2

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

    D=112−4⋅(−1)⋅(−10)=121−40=81D = 11^2 - 4\cdot(-1)\cdot(-10) = 121 - 40 = 81
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(x−1)(x−10)=0-\left(x - 1\right)\left(x - 10\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x−1=0⇒x=1x−10=0⇒x=10x - 1 = 0 \Rightarrow x = 1\qquad x - 10 = 0 \Rightarrow x = 10
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±81−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{81}}{-2}
  7. 7

    Simplify to get the solutions:

    x=1orx=10x = 1\quad\text{or}\quad x = 10
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = -1·(1) + 11·(1) + (-10) = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 11x - 10 = 0\ \Rightarrow\ x = 1,\ 10 \]
\[ -x^2 + 11x - 10 = 0 \]
\[ D = 11^2 - 4\cdot(-1)\cdot(-10) = 121 - 40 = 81 \]
\[ -\left(x - 1\right)\left(x - 10\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x - 10 = 0 \Rightarrow x = 10 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{81}}{-2} \]
\[ x = 1\quad\text{or}\quad x = 10 \]

The parabola y = -x² + 11x - 10

xy
2.511.25
3.516.25
4.519.25
5.520.25
6.519.25
7.516.25
8.511.25

Questions

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

The solutions of -x² + 11x - 10 = 0 are x = 1 or x = 10.

Does -x² + 11x - 10 = 0 factor?

Yes: -1(x-1)(x-10) = 0.

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

(11/2, 81/4). The parabola opens downwards, so this is the maximum.

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 = 1 or x = 10 (steps) | WordGPT