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Solve -x² + 7x - 11 = 0

The solutions of -x² + 7x - 11 = 0 are x = (7 + √5)/2 or x = (7 - √5)/2, that is x ≈ 2.381966 and x ≈ 4.618034.

x² +x += 0
−x2+7x−11=0⇒x=7+52, 7−52-x^2 + 7x - 11 = 0 \Rightarrow x = \frac{7 + \sqrt{5}}{2},\ \frac{7 - \sqrt{5}}{2}
Discriminant D = 5 · x ≈ 2.38197, 4.61803

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

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Graph of y = -x² + 7x - 111234567-12-10-8-6-4-22x = 2.381966x = 4.618034vertexy = -x² + 7x - 11
Graph of y = -x² + 7x - 11. The curve crosses the x-axis at x = 2.381966 and x = 4.618034; the vertex is at (3.5, 1.25).

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 = 7, c = -11.

    −x2+7x−11=0-x^2 + 7x - 11 = 0
  2. 2

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

    D=72−4⋅(−1)⋅(−11)=49−44=5D = 7^2 - 4\cdot(-1)\cdot(-11) = 49 - 44 = 5
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−7±5−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{5}}{-2}
  5. 5

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

    x=7+52orx=7−52x = \frac{7 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{5}}{2}
  6. 6

    As decimals: x ≈ 2.381966 and x ≈ 4.618034.

  7. 7

    Check numerically: for x ≈ 2.381966 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 7x - 11 = 0\ \Rightarrow\ x = \frac{7 + \sqrt{5}}{2},\ \frac{7 - \sqrt{5}}{2} \]
\[ -x^2 + 7x - 11 = 0 \]
\[ D = 7^2 - 4\cdot(-1)\cdot(-11) = 49 - 44 = 5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{5}}{-2} \]
\[ x = \frac{7 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{5}}{2} \]

The parabola y = -x² + 7x - 11

xy
0.5-7.75
1.5-2.75
2.50.25
3.51.25
4.50.25
5.5-2.75
6.5-7.75

Questions

What are the solutions of -x² + 7x - 11 = 0?

The solutions of -x² + 7x - 11 = 0 are x = (7 + √5)/2 or x = (7 - √5)/2, that is x ≈ 2.381966 and x ≈ 4.618034.

Does -x² + 7x - 11 = 0 factor?

Not over the integers, because the discriminant 5 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 7x - 11?

(7/2, 5/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 7, product = c/a = 11.

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    Solve -x² + 7x - 11 = 0: x = (7 + √5)/2 or x = (7 - √5)/2 (steps) | WordGPT