WordGPT

Solve x² - 8x - 1 = 0

The solutions of x² - 8x - 1 = 0 are x = 4 + √17 or x = 4 - √17, that is x ≈ 8.1231056 and x ≈ -0.12310563.

x² +x += 0
x2−8x−1=0⇒x=4+17, 4−17x^2 - 8x - 1 = 0 \Rightarrow x = 4 + \sqrt{17},\ 4 - \sqrt{17}
Discriminant D = 68 · x ≈ 8.12311, -0.123106

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

Explain it step by step in chatMake a short video of it
Graph of y = x² - 8x - 1-2246810-20-10102030x = 8.1231056x = -0.12310563vertexy = x² - 8x - 1
Graph of y = x² - 8x - 1. The curve crosses the x-axis at x = 8.1231056 and x = -0.12310563; the vertex is at (4, -17).

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 = -1.

    x2−8x−1=0x^2 - 8x - 1 = 0
  2. 2

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

    D=(−8)2−4⋅1⋅(−1)=64+4=68D = (-8)^2 - 4\cdot1\cdot(-1) = 64 + 4 = 68
  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±682x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{68}}{2}
  5. 5

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

    x=4+17orx=4−17x = 4 + \sqrt{17}\quad\text{or}\quad x = 4 - \sqrt{17}
  6. 6

    As decimals: x ≈ 8.1231056 and x ≈ -0.12310563.

  7. 7

    Check numerically: for x ≈ 8.1231056 the left side is 1.4210855e-14 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 8x - 1 = 0\ \Rightarrow\ x = 4 + \sqrt{17},\ 4 - \sqrt{17} \]
\[ x^2 - 8x - 1 = 0 \]
\[ D = (-8)^2 - 4\cdot1\cdot(-1) = 64 + 4 = 68 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{68}}{2} \]
\[ x = 4 + \sqrt{17}\quad\text{or}\quad x = 4 - \sqrt{17} \]

The parabola y = x² - 8x - 1

xy
1-8
2-13
3-16
4-17
5-16
6-13
7-8

Questions

What are the solutions of x² - 8x - 1 = 0?

The solutions of x² - 8x - 1 = 0 are x = 4 + √17 or x = 4 - √17, that is x ≈ 8.1231056 and x ≈ -0.12310563.

Does x² - 8x - 1 = 0 factor?

Not over the integers, because the discriminant 68 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 8x - 1?

(4, -17). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = -1.

Similar equations

    Solve x² - 8x - 1 = 0: x = 4 + √17 or x = 4 - √17 (steps) | WordGPT