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Solve x² + 12x + 11 = 0

The solutions of x² + 12x + 11 = 0 are x = -1 or x = -11.

x² +x += 0
x2+12x+11=0⇒x=−1, −11x^2 + 12x + 11 = 0 \Rightarrow x = -1,\ -11
Discriminant D = 100 · 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 + 12x + 11-12-10-8-6-4-2-30-20-10102030x = -1x = -11vertexy = x² + 12x + 11
Graph of y = x² + 12x + 11. The curve crosses the x-axis at x = -1 and x = -11; the vertex is at (-6, -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 = 12, c = 11.

    x2+12x+11=0x^2 + 12x + 11 = 0
  2. 2

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

    D=122−4⋅1⋅11=144−44=100D = 12^2 - 4\cdot1\cdot11 = 144 - 44 = 100
  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=−12±1002x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{100}}{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) + 12·(-1) + 11 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 12x + 11 = 0\ \Rightarrow\ x = -1,\ -11 \]
\[ x^2 + 12x + 11 = 0 \]
\[ D = 12^2 - 4\cdot1\cdot11 = 144 - 44 = 100 \]
\[ \left(x + 1\right)\left(x + 11\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x + 11 = 0 \Rightarrow x = -11 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{100}}{2} \]
\[ x = -1\quad\text{or}\quad x = -11 \]

The parabola y = x² + 12x + 11

xy
-9-16
-8-21
-7-24
-6-25
-5-24
-4-21
-3-16

Questions

What are the solutions of x² + 12x + 11 = 0?

The solutions of x² + 12x + 11 = 0 are x = -1 or x = -11.

Does x² + 12x + 11 = 0 factor?

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

What is the vertex of y = x² + 12x + 11?

(-6, -25). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -12, product = c/a = 11.

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