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Solve -x² + 10x - 5 = 0

The solutions of -x² + 10x - 5 = 0 are x = 5 + 2√5 or x = 5 - 2√5, that is x ≈ 0.52786405 and x ≈ 9.472136.

x² +x += 0
−x2+10x−5=0⇒x=5+25, 5−25-x^2 + 10x - 5 = 0 \Rightarrow x = 5 + 2\sqrt{5},\ 5 - 2\sqrt{5}
Discriminant D = 80 · x ≈ 0.527864, 9.47214

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

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Graph of y = -x² + 10x - 5246810-30-20-101020x = 0.52786405x = 9.472136vertexy = -x² + 10x - 5
Graph of y = -x² + 10x - 5. The curve crosses the x-axis at x = 0.52786405 and x = 9.472136; the vertex is at (5, 20).

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=80D = 10^2 - 4\cdot(-1)\cdot(-5) = 100 - 20 = 80
  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±80−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{80}}{-2}
  5. 5

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

    x=5+25orx=5−25x = 5 + 2\sqrt{5}\quad\text{or}\quad x = 5 - 2\sqrt{5}
  6. 6

    As decimals: x ≈ 0.52786405 and x ≈ 9.472136.

  7. 7

    Check numerically: for x ≈ 0.52786405 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 10x - 5 = 0\ \Rightarrow\ x = 5 + 2\sqrt{5},\ 5 - 2\sqrt{5} \]
\[ -x^2 + 10x - 5 = 0 \]
\[ D = 10^2 - 4\cdot(-1)\cdot(-5) = 100 - 20 = 80 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{80}}{-2} \]
\[ x = 5 + 2\sqrt{5}\quad\text{or}\quad x = 5 - 2\sqrt{5} \]

The parabola y = -x² + 10x - 5

xy
211
316
419
520
619
716
811

Questions

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

The solutions of -x² + 10x - 5 = 0 are x = 5 + 2√5 or x = 5 - 2√5, that is x ≈ 0.52786405 and x ≈ 9.472136.

Does -x² + 10x - 5 = 0 factor?

Not over the integers, because the discriminant 80 is not a perfect square. Use the quadratic formula.

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

(5, 20). 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 + 2√5 or x = 5 - 2√5 (steps) | WordGPT