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Solve -x² - 10x + 7 = 0

The solutions of -x² - 10x + 7 = 0 are x = -5 + 4√2 or x = -5 - 4√2, that is x ≈ -10.656854 and x ≈ 0.65685425.

x² +x += 0
−x2−10x+7=0⇒x=−5+42, −5−42-x^2 - 10x + 7 = 0 \Rightarrow x = -5 + 4\sqrt{2},\ -5 - 4\sqrt{2}
Discriminant D = 128 · x ≈ -10.6569, 0.656854

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

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Graph of y = -x² - 10x + 7-12-10-8-6-4-22-30-20-1010203040x = -10.656854x = 0.65685425vertexy = -x² - 10x + 7
Graph of y = -x² - 10x + 7. The curve crosses the x-axis at x = -10.656854 and x = 0.65685425; the vertex is at (-5, 32).

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 = 7.

    −x2−10x+7=0-x^2 - 10x + 7 = 0
  2. 2

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

    D=(−10)2−4⋅(−1)⋅7=100+28=128D = (-10)^2 - 4\cdot(-1)\cdot7 = 100 + 28 = 128
  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±128−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{128}}{-2}
  5. 5

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

    x=−5+42orx=−5−42x = -5 + 4\sqrt{2}\quad\text{or}\quad x = -5 - 4\sqrt{2}
  6. 6

    As decimals: x ≈ -10.656854 and x ≈ 0.65685425.

  7. 7

    Check numerically: for x ≈ -10.656854 the left side is 1.4210855e-14 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 - 10x + 7 = 0\ \Rightarrow\ x = -5 + 4\sqrt{2},\ -5 - 4\sqrt{2} \]
\[ -x^2 - 10x + 7 = 0 \]
\[ D = (-10)^2 - 4\cdot(-1)\cdot7 = 100 + 28 = 128 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{128}}{-2} \]
\[ x = -5 + 4\sqrt{2}\quad\text{or}\quad x = -5 - 4\sqrt{2} \]

The parabola y = -x² - 10x + 7

xy
-823
-728
-631
-532
-431
-328
-223

Questions

What are the solutions of -x² - 10x + 7 = 0?

The solutions of -x² - 10x + 7 = 0 are x = -5 + 4√2 or x = -5 - 4√2, that is x ≈ -10.656854 and x ≈ 0.65685425.

Does -x² - 10x + 7 = 0 factor?

Not over the integers, because the discriminant 128 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² - 10x + 7?

(-5, 32). 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 = -7.

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    Solve -x² - 10x + 7 = 0: x = -5 + 4√2 or x = -5 - 4√2 (steps) | WordGPT