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Solve -x² + 8x - 10 = 0

The solutions of -x² + 8x - 10 = 0 are x = 4 + √6 or x = 4 - √6, that is x ≈ 1.5505103 and x ≈ 6.4494897.

x² +x += 0
−x2+8x−10=0⇒x=4+6, 4−6-x^2 + 8x - 10 = 0 \Rightarrow x = 4 + \sqrt{6},\ 4 - \sqrt{6}
Discriminant D = 24 · x ≈ 1.55051, 6.44949

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

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Graph of y = -x² + 8x - 1012345678-20-15-10-55x = 1.5505103x = 6.4494897vertexy = -x² + 8x - 10
Graph of y = -x² + 8x - 10. The curve crosses the x-axis at x = 1.5505103 and x = 6.4494897; the vertex is at (4, 6).

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 = -10.

    −x2+8x−10=0-x^2 + 8x - 10 = 0
  2. 2

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

    D=82−4⋅(−1)⋅(−10)=64−40=24D = 8^2 - 4\cdot(-1)\cdot(-10) = 64 - 40 = 24
  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±24−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{24}}{-2}
  5. 5

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

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

    As decimals: x ≈ 1.5505103 and x ≈ 6.4494897.

  7. 7

    Check numerically: for x ≈ 1.5505103 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 8x - 10 = 0\ \Rightarrow\ x = 4 + \sqrt{6},\ 4 - \sqrt{6} \]
\[ -x^2 + 8x - 10 = 0 \]
\[ D = 8^2 - 4\cdot(-1)\cdot(-10) = 64 - 40 = 24 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{24}}{-2} \]
\[ x = 4 + \sqrt{6}\quad\text{or}\quad x = 4 - \sqrt{6} \]

The parabola y = -x² + 8x - 10

xy
1-3
22
35
46
55
62
7-3

Questions

What are the solutions of -x² + 8x - 10 = 0?

The solutions of -x² + 8x - 10 = 0 are x = 4 + √6 or x = 4 - √6, that is x ≈ 1.5505103 and x ≈ 6.4494897.

Does -x² + 8x - 10 = 0 factor?

Not over the integers, because the discriminant 24 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 8x - 10?

(4, 6). 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 = 10.

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    Solve -x² + 8x - 10 = 0: x = 4 + √6 or x = 4 - √6 (steps) | WordGPT