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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, 11-x^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 - 1124681012-30-20-10102030x = 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=0-x^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\cdot(-1)\cdot(-11) = 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±100−2x = \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
\[ -x^2 + 12x - 11 = 0\ \Rightarrow\ x = 1,\ 11 \]
\[ -x^2 + 12x - 11 = 0 \]
\[ D = 12^2 - 4\cdot(-1)\cdot(-11) = 144 - 44 = 100 \]
\[ -\left(x - 1\right)\left(x - 11\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x - 11 = 0 \Rightarrow x = 11 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{100}}{-2} \]
\[ x = 1\quad\text{or}\quad x = 11 \]

The parabola y = -x² + 12x - 11

xy
316
421
524
625
724
821
916

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

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

(6, 25). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 12, product = c/a = 11.

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