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Solve -2x² + 12x - 11 = 0

The solutions of -2x² + 12x - 11 = 0 are x = (6 + √14)/2 or x = (6 - √14)/2, that is x ≈ 1.1291713 and x ≈ 4.8708287.

x² +x += 0
−2x2+12x−11=0⇒x=6+142, 6−142-2x^2 + 12x - 11 = 0 \Rightarrow x = \frac{6 + \sqrt{14}}{2},\ \frac{6 - \sqrt{14}}{2}
Discriminant D = 56 · x ≈ 1.12917, 4.87083

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

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Graph of y = -2x² + 12x - 11-11234567-30-20-1010x = 1.1291713x = 4.8708287vertexy = -2x² + 12x - 11
Graph of y = -2x² + 12x - 11. The curve crosses the x-axis at x = 1.1291713 and x = 4.8708287; the vertex is at (3, 7).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = 12, c = -11.

    −2x2+12x−11=0-2x^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⋅(−2)⋅(−11)=144−88=56D = 12^2 - 4\cdot(-2)\cdot(-11) = 144 - 88 = 56
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−12±56−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{56}}{-4}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=6+142orx=6−142x = \frac{6 + \sqrt{14}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{14}}{2}
  6. 6

    As decimals: x ≈ 1.1291713 and x ≈ 4.8708287.

  7. 7

    Check numerically: for x ≈ 1.1291713 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 12x - 11 = 0\ \Rightarrow\ x = \frac{6 + \sqrt{14}}{2},\ \frac{6 - \sqrt{14}}{2} \]
\[ -2x^2 + 12x - 11 = 0 \]
\[ D = 12^2 - 4\cdot(-2)\cdot(-11) = 144 - 88 = 56 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{56}}{-4} \]
\[ x = \frac{6 + \sqrt{14}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{14}}{2} \]

The parabola y = -2x² + 12x - 11

xy
0-11
1-1
25
37
45
5-1
6-11

Questions

What are the solutions of -2x² + 12x - 11 = 0?

The solutions of -2x² + 12x - 11 = 0 are x = (6 + √14)/2 or x = (6 - √14)/2, that is x ≈ 1.1291713 and x ≈ 4.8708287.

Does -2x² + 12x - 11 = 0 factor?

Not over the integers, because the discriminant 56 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² + 12x - 11?

(3, 7). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = 11/2.

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    Solve -2x² + 12x - 11 = 0: x = (6 + √14)/2 or x = (6 - √14)/2 (steps) | WordGPT