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Solve x² + 10x - 11 = 0

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

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

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

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Graph of y = x² + 10x - 11-12-10-8-6-4-22-40-30-20-1010203040x = 1x = -11vertexy = x² + 10x - 11
Graph of y = x² + 10x - 11. The curve crosses the x-axis at x = 1 and x = -11; 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=102−4⋅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−1)(x+11)=0\left(x - 1\right)\left(x + 11\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+11=0⇒x=−11x - 1 = 0 \Rightarrow x = 1\qquad x + 11 = 0 \Rightarrow x = -11
  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=1orx=−11x = 1\quad\text{or}\quad x = -11
  8. 8

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

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

The parabola y = x² + 10x - 11

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

Questions

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

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

Does x² + 10x - 11 = 0 factor?

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