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Solve x² - 7x - 7 = 0

The solutions of x² - 7x - 7 = 0 are x = (7 + √77)/2 or x = (7 - √77)/2, that is x ≈ 7.8874822 and x ≈ -0.88748219.

x² +x += 0
x2−7x−7=0⇒x=7+772, 7−772x^2 - 7x - 7 = 0 \Rightarrow x = \frac{7 + \sqrt{77}}{2},\ \frac{7 - \sqrt{77}}{2}
Discriminant D = 77 · x ≈ 7.88748, -0.887482

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

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Graph of y = x² - 7x - 7-2246810-20-10102030x = 7.8874822x = -0.88748219vertexy = x² - 7x - 7
Graph of y = x² - 7x - 7. The curve crosses the x-axis at x = 7.8874822 and x = -0.88748219; the vertex is at (3.5, -19.25).

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 = -7, c = -7.

    x2−7x−7=0x^2 - 7x - 7 = 0
  2. 2

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

    D=(−7)2−4⋅1⋅(−7)=49+28=77D = (-7)^2 - 4\cdot1\cdot(-7) = 49 + 28 = 77
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=7±772x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{77}}{2}
  5. 5

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

    x=7+772orx=7−772x = \frac{7 + \sqrt{77}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{77}}{2}
  6. 6

    As decimals: x ≈ 7.8874822 and x ≈ -0.88748219.

  7. 7

    Check numerically: for x ≈ 7.8874822 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 7x - 7 = 0\ \Rightarrow\ x = \frac{7 + \sqrt{77}}{2},\ \frac{7 - \sqrt{77}}{2} \]
\[ x^2 - 7x - 7 = 0 \]
\[ D = (-7)^2 - 4\cdot1\cdot(-7) = 49 + 28 = 77 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{77}}{2} \]
\[ x = \frac{7 + \sqrt{77}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{77}}{2} \]

The parabola y = x² - 7x - 7

xy
0.5-10.25
1.5-15.25
2.5-18.25
3.5-19.25
4.5-18.25
5.5-15.25
6.5-10.25

Questions

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

The solutions of x² - 7x - 7 = 0 are x = (7 + √77)/2 or x = (7 - √77)/2, that is x ≈ 7.8874822 and x ≈ -0.88748219.

Does x² - 7x - 7 = 0 factor?

Not over the integers, because the discriminant 77 is not a perfect square. Use the quadratic formula.

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

(7/2, -77/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 7, product = c/a = -7.

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    Solve x² - 7x - 7 = 0: x = (7 + √77)/2 or x = (7 - √77)/2 (steps) | WordGPT