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Solve x² - 10x - 11 = 0

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

x² +x += 0
x2−10x−11=0⇒x=11, −1x^2 - 10x - 11 = 0 \Rightarrow x = 11,\ -1
Discriminant D = 144 · x ≈ 11, -1

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

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Graph of y = x² - 10x - 11-224681012-40-30-20-1010203040x = 11x = -1vertexy = x² - 10x - 11
Graph of y = x² - 10x - 11. The curve crosses the x-axis at x = 11 and x = -1; the vertex is at (5, -36).

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 = -10, c = -11.

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

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

    D=(−10)2−4⋅1⋅(−11)=100+44=144D = (-10)^2 - 4\cdot1\cdot(-11) = 100 + 44 = 144
  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−11)(x+1)=0\left(x - 11\right)\left(x + 1\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−11=0⇒x=11x+1=0⇒x=−1x - 11 = 0 \Rightarrow x = 11\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±1442x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{144}}{2}
  7. 7

    Simplify to get the solutions:

    x=11orx=−1x = 11\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 11 back in: a·x² + b·x + c = 1·(121) + (-10)·(11) + (-11) = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - 10x - 11 = 0\ \Rightarrow\ x = 11,\ -1 \]
\[ x^2 - 10x - 11 = 0 \]
\[ D = (-10)^2 - 4\cdot1\cdot(-11) = 100 + 44 = 144 \]
\[ \left(x - 11\right)\left(x + 1\right) = 0 \]
\[ x - 11 = 0 \Rightarrow x = 11\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{144}}{2} \]
\[ x = 11\quad\text{or}\quad x = -1 \]

The parabola y = x² - 10x - 11

xy
2-27
3-32
4-35
5-36
6-35
7-32
8-27

Questions

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

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

Does x² - 10x - 11 = 0 factor?

Yes: (x-11)(x+1) = 0.

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

(5, -36). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 10, product = c/a = -11.

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