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Solve -2x² + 10x - 10 = 0

The solutions of -2x² + 10x - 10 = 0 are x = (5 + √5)/2 or x = (5 - √5)/2, that is x ≈ 1.381966 and x ≈ 3.618034.

x² +x += 0
−2x2+10x−10=0⇒x=5+52, 5−52-2x^2 + 10x - 10 = 0 \Rightarrow x = \frac{5 + \sqrt{5}}{2},\ \frac{5 - \sqrt{5}}{2}
Discriminant D = 20 · x ≈ 1.38197, 3.61803

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

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Graph of y = -2x² + 10x - 10-1123456-25-20-15-10-55x = 1.381966x = 3.618034vertexy = -2x² + 10x - 10
Graph of y = -2x² + 10x - 10. The curve crosses the x-axis at x = 1.381966 and x = 3.618034; the vertex is at (2.5, 2.5).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = 10, c = -10.

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

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

    D=102−4⋅(−2)⋅(−10)=100−80=20D = 10^2 - 4\cdot(-2)\cdot(-10) = 100 - 80 = 20
  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±20−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{20}}{-4}
  5. 5

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

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

    As decimals: x ≈ 1.381966 and x ≈ 3.618034.

  7. 7

    Check numerically: for x ≈ 1.381966 the left side is 0 (zero up to rounding) ✓

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

The parabola y = -2x² + 10x - 10

xy
-0.5-15.5
0.5-5.5
1.50.5
2.52.5
3.50.5
4.5-5.5
5.5-15.5

Questions

What are the solutions of -2x² + 10x - 10 = 0?

The solutions of -2x² + 10x - 10 = 0 are x = (5 + √5)/2 or x = (5 - √5)/2, that is x ≈ 1.381966 and x ≈ 3.618034.

Does -2x² + 10x - 10 = 0 factor?

Not over the integers, because the discriminant 20 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² + 10x - 10?

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

What are the sum and product of the roots?

Sum = −b/a = 5, product = c/a = 5.

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