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Solve x² - 7x - 1 = 0

The solutions of x² - 7x - 1 = 0 are x = (7 + √53)/2 or x = (7 - √53)/2, that is x ≈ 7.1400549 and x ≈ -0.14005494.

x² +x += 0
x2−7x−1=0⇒x=7+532, 7−532x^2 - 7x - 1 = 0 \Rightarrow x = \frac{7 + \sqrt{53}}{2},\ \frac{7 - \sqrt{53}}{2}
Discriminant D = 53 · x ≈ 7.14005, -0.140055

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

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Graph of y = x² - 7x - 1-22468-101020x = 7.1400549x = -0.14005494vertexy = x² - 7x - 1
Graph of y = x² - 7x - 1. The curve crosses the x-axis at x = 7.1400549 and x = -0.14005494; the vertex is at (3.5, -13.25).

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 = -1.

    x2−7x−1=0x^2 - 7x - 1 = 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⋅(−1)=49+4=53D = (-7)^2 - 4\cdot1\cdot(-1) = 49 + 4 = 53
  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±532x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{53}}{2}
  5. 5

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

    x=7+532orx=7−532x = \frac{7 + \sqrt{53}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{53}}{2}
  6. 6

    As decimals: x ≈ 7.1400549 and x ≈ -0.14005494.

  7. 7

    Check numerically: for x ≈ 7.1400549 the left side is -7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 7x - 1 = 0\ \Rightarrow\ x = \frac{7 + \sqrt{53}}{2},\ \frac{7 - \sqrt{53}}{2} \]
\[ x^2 - 7x - 1 = 0 \]
\[ D = (-7)^2 - 4\cdot1\cdot(-1) = 49 + 4 = 53 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{53}}{2} \]
\[ x = \frac{7 + \sqrt{53}}{2}\quad\text{or}\quad x = \frac{7 - \sqrt{53}}{2} \]

The parabola y = x² - 7x - 1

xy
0.5-4.25
1.5-9.25
2.5-12.25
3.5-13.25
4.5-12.25
5.5-9.25
6.5-4.25

Questions

What are the solutions of x² - 7x - 1 = 0?

The solutions of x² - 7x - 1 = 0 are x = (7 + √53)/2 or x = (7 - √53)/2, that is x ≈ 7.1400549 and x ≈ -0.14005494.

Does x² - 7x - 1 = 0 factor?

Not over the integers, because the discriminant 53 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 7x - 1?

(7/2, -53/4). 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 = -1.

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    Solve x² - 7x - 1 = 0: x = (7 + √53)/2 or x = (7 - √53)/2 (steps) | WordGPT