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Solve -x² + 8x + 1 = 0

The solutions of -x² + 8x + 1 = 0 are x = 4 + √17 or x = 4 - √17, that is x ≈ -0.12310563 and x ≈ 8.1231056.

x² +x += 0
−x2+8x+1=0⇒x=4+17, 4−17-x^2 + 8x + 1 = 0 \Rightarrow x = 4 + \sqrt{17},\ 4 - \sqrt{17}
Discriminant D = 68 · x ≈ -0.123106, 8.12311

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

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Graph of y = -x² + 8x + 1-2246810-30-20-101020x = -0.12310563x = 8.1231056vertexy = -x² + 8x + 1
Graph of y = -x² + 8x + 1. The curve crosses the x-axis at x = -0.12310563 and x = 8.1231056; the vertex is at (4, 17).

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 = 8, c = 1.

    −x2+8x+1=0-x^2 + 8x + 1 = 0
  2. 2

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

    D=82−4⋅(−1)⋅1=64+4=68D = 8^2 - 4\cdot(-1)\cdot1 = 64 + 4 = 68
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−8±68−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{68}}{-2}
  5. 5

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

    x=4+17orx=4−17x = 4 + \sqrt{17}\quad\text{or}\quad x = 4 - \sqrt{17}
  6. 6

    As decimals: x ≈ -0.12310563 and x ≈ 8.1231056.

  7. 7

    Check numerically: for x ≈ -0.12310563 the left side is -2.220446e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 8x + 1 = 0\ \Rightarrow\ x = 4 + \sqrt{17},\ 4 - \sqrt{17} \]
\[ -x^2 + 8x + 1 = 0 \]
\[ D = 8^2 - 4\cdot(-1)\cdot1 = 64 + 4 = 68 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{68}}{-2} \]
\[ x = 4 + \sqrt{17}\quad\text{or}\quad x = 4 - \sqrt{17} \]

The parabola y = -x² + 8x + 1

xy
18
213
316
417
516
613
78

Questions

What are the solutions of -x² + 8x + 1 = 0?

The solutions of -x² + 8x + 1 = 0 are x = 4 + √17 or x = 4 - √17, that is x ≈ -0.12310563 and x ≈ 8.1231056.

Does -x² + 8x + 1 = 0 factor?

Not over the integers, because the discriminant 68 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 8x + 1?

(4, 17). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = -1.

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    Solve -x² + 8x + 1 = 0: x = 4 + √17 or x = 4 - √17 (steps) | WordGPT