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Solve -x² + 10x + 5 = 0

The solutions of -x² + 10x + 5 = 0 are x = 5 + √30 or x = 5 - √30, that is x ≈ -0.47722558 and x ≈ 10.477226.

x² +x += 0
−x2+10x+5=0⇒x=5+30, 5−30-x^2 + 10x + 5 = 0 \Rightarrow x = 5 + \sqrt{30},\ 5 - \sqrt{30}
Discriminant D = 120 · x ≈ -0.477226, 10.4772

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

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Graph of y = -x² + 10x + 5-224681012-30-20-10102030x = -0.47722558x = 10.477226vertexy = -x² + 10x + 5
Graph of y = -x² + 10x + 5. The curve crosses the x-axis at x = -0.47722558 and x = 10.477226; the vertex is at (5, 30).

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 = 10, c = 5.

    −x2+10x+5=0-x^2 + 10x + 5 = 0
  2. 2

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

    D=102−4⋅(−1)⋅5=100+20=120D = 10^2 - 4\cdot(-1)\cdot5 = 100 + 20 = 120
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−10±120−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{120}}{-2}
  5. 5

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

    x=5+30orx=5−30x = 5 + \sqrt{30}\quad\text{or}\quad x = 5 - \sqrt{30}
  6. 6

    As decimals: x ≈ -0.47722558 and x ≈ 10.477226.

  7. 7

    Check numerically: for x ≈ -0.47722558 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 10x + 5 = 0\ \Rightarrow\ x = 5 + \sqrt{30},\ 5 - \sqrt{30} \]
\[ -x^2 + 10x + 5 = 0 \]
\[ D = 10^2 - 4\cdot(-1)\cdot5 = 100 + 20 = 120 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{120}}{-2} \]
\[ x = 5 + \sqrt{30}\quad\text{or}\quad x = 5 - \sqrt{30} \]

The parabola y = -x² + 10x + 5

xy
221
326
429
530
629
726
821

Questions

What are the solutions of -x² + 10x + 5 = 0?

The solutions of -x² + 10x + 5 = 0 are x = 5 + √30 or x = 5 - √30, that is x ≈ -0.47722558 and x ≈ 10.477226.

Does -x² + 10x + 5 = 0 factor?

Not over the integers, because the discriminant 120 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 10x + 5?

(5, 30). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 10, product = c/a = -5.

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    Solve -x² + 10x + 5 = 0: x = 5 + √30 or x = 5 - √30 (steps) | WordGPT