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Solve -x² + 8x + 7 = 0

The solutions of -x² + 8x + 7 = 0 are x = 4 + √23 or x = 4 - √23, that is x ≈ -0.79583152 and x ≈ 8.7958315.

x² +x += 0
−x2+8x+7=0⇒x=4+23, 4−23-x^2 + 8x + 7 = 0 \Rightarrow x = 4 + \sqrt{23},\ 4 - \sqrt{23}
Discriminant D = 92 · x ≈ -0.795832, 8.79583

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

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Graph of y = -x² + 8x + 7-2246810-30-20-10102030x = -0.79583152x = 8.7958315vertexy = -x² + 8x + 7
Graph of y = -x² + 8x + 7. The curve crosses the x-axis at x = -0.79583152 and x = 8.7958315; the vertex is at (4, 23).

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 = 7.

    −x2+8x+7=0-x^2 + 8x + 7 = 0
  2. 2

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

    D=82−4⋅(−1)⋅7=64+28=92D = 8^2 - 4\cdot(-1)\cdot7 = 64 + 28 = 92
  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±92−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{92}}{-2}
  5. 5

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

    x=4+23orx=4−23x = 4 + \sqrt{23}\quad\text{or}\quad x = 4 - \sqrt{23}
  6. 6

    As decimals: x ≈ -0.79583152 and x ≈ 8.7958315.

  7. 7

    Check numerically: for x ≈ -0.79583152 the left side is 4.4408921e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 8x + 7 = 0\ \Rightarrow\ x = 4 + \sqrt{23},\ 4 - \sqrt{23} \]
\[ -x^2 + 8x + 7 = 0 \]
\[ D = 8^2 - 4\cdot(-1)\cdot7 = 64 + 28 = 92 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{92}}{-2} \]
\[ x = 4 + \sqrt{23}\quad\text{or}\quad x = 4 - \sqrt{23} \]

The parabola y = -x² + 8x + 7

xy
114
219
322
423
522
619
714

Questions

What are the solutions of -x² + 8x + 7 = 0?

The solutions of -x² + 8x + 7 = 0 are x = 4 + √23 or x = 4 - √23, that is x ≈ -0.79583152 and x ≈ 8.7958315.

Does -x² + 8x + 7 = 0 factor?

Not over the integers, because the discriminant 92 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 8x + 7?

(4, 23). 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 = -7.

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    Solve -x² + 8x + 7 = 0: x = 4 + √23 or x = 4 - √23 (steps) | WordGPT