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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, −10x^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.

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Graph of y = x² + 11x + 10-12-10-8-6-4-2-20-10102030x = -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=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=112−4⋅1⋅10=121−40=81D = 11^2 - 4\cdot1\cdot10 = 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±812x = \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
\[ x^2 + 11x + 10 = 0\ \Rightarrow\ x = -1,\ -10 \]
\[ x^2 + 11x + 10 = 0 \]
\[ D = 11^2 - 4\cdot1\cdot10 = 121 - 40 = 81 \]
\[ \left(x + 1\right)\left(x + 10\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x + 10 = 0 \Rightarrow x = -10 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{81}}{2} \]
\[ x = -1\quad\text{or}\quad x = -10 \]

The parabola y = x² + 11x + 10

xy
-8.5-11.25
-7.5-16.25
-6.5-19.25
-5.5-20.25
-4.5-19.25
-3.5-16.25
-2.5-11.25

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: (x+1)(x+10) = 0.

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

(-11/2, -81/4). 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.

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