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Solve x² - 12x + 11 = 0

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

x² +x += 0
x2−12x+11=0⇒x=11, 1x^2 - 12x + 11 = 0 \Rightarrow x = 11,\ 1
Discriminant D = 100 · 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 - 12x + 1124681012-30-20-10102030x = 11x = 1vertexy = x² - 12x + 11
Graph of y = x² - 12x + 11. The curve crosses the x-axis at x = 11 and x = 1; 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=(−12)2−4⋅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−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=12±1002x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{100}}{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) + (-12)·(11) + 11 = 0 ✓

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

The parabola y = x² - 12x + 11

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

Questions

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

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

Does x² - 12x + 11 = 0 factor?

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