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Solve -x² + 10x + 2 = 0

The solutions of -x² + 10x + 2 = 0 are x = 5 + 3√3 or x = 5 - 3√3, that is x ≈ -0.19615242 and x ≈ 10.196152.

x² +x += 0
−x2+10x+2=0⇒x=5+33, 5−33-x^2 + 10x + 2 = 0 \Rightarrow x = 5 + 3\sqrt{3},\ 5 - 3\sqrt{3}
Discriminant D = 108 · x ≈ -0.196152, 10.1962

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

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Graph of y = -x² + 10x + 2-224681012-30-20-10102030x = -0.19615242x = 10.196152vertexy = -x² + 10x + 2
Graph of y = -x² + 10x + 2. The curve crosses the x-axis at x = -0.19615242 and x = 10.196152; the vertex is at (5, 27).

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 = 2.

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

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

    D=102−4⋅(−1)⋅2=100+8=108D = 10^2 - 4\cdot(-1)\cdot2 = 100 + 8 = 108
  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±108−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{108}}{-2}
  5. 5

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

    x=5+33orx=5−33x = 5 + 3\sqrt{3}\quad\text{or}\quad x = 5 - 3\sqrt{3}
  6. 6

    As decimals: x ≈ -0.19615242 and x ≈ 10.196152.

  7. 7

    Check numerically: for x ≈ -0.19615242 the left side is -1.3322676e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 10x + 2 = 0\ \Rightarrow\ x = 5 + 3\sqrt{3},\ 5 - 3\sqrt{3} \]
\[ -x^2 + 10x + 2 = 0 \]
\[ D = 10^2 - 4\cdot(-1)\cdot2 = 100 + 8 = 108 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{108}}{-2} \]
\[ x = 5 + 3\sqrt{3}\quad\text{or}\quad x = 5 - 3\sqrt{3} \]

The parabola y = -x² + 10x + 2

xy
218
323
426
527
626
723
818

Questions

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

The solutions of -x² + 10x + 2 = 0 are x = 5 + 3√3 or x = 5 - 3√3, that is x ≈ -0.19615242 and x ≈ 10.196152.

Does -x² + 10x + 2 = 0 factor?

Not over the integers, because the discriminant 108 is not a perfect square. Use the quadratic formula.

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

(5, 27). 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 = -2.

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    Solve -x² + 10x + 2 = 0: x = 5 + 3√3 or x = 5 - 3√3 (steps) | WordGPT