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Solve x² - 8x - 4 = 0

The solutions of x² - 8x - 4 = 0 are x = 4 + 2√5 or x = 4 - 2√5, that is x ≈ 8.472136 and x ≈ -0.47213595.

x² +x += 0
x2−8x−4=0⇒x=4+25, 4−25x^2 - 8x - 4 = 0 \Rightarrow x = 4 + 2\sqrt{5},\ 4 - 2\sqrt{5}
Discriminant D = 80 · x ≈ 8.47214, -0.472136

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

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Graph of y = x² - 8x - 4-2246810-20-10102030x = 8.472136x = -0.47213595vertexy = x² - 8x - 4
Graph of y = x² - 8x - 4. The curve crosses the x-axis at x = 8.472136 and x = -0.47213595; the vertex is at (4, -20).

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 = -4.

    x2−8x−4=0x^2 - 8x - 4 = 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⋅(−4)=64+16=80D = (-8)^2 - 4\cdot1\cdot(-4) = 64 + 16 = 80
  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±802x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{80}}{2}
  5. 5

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

    x=4+25orx=4−25x = 4 + 2\sqrt{5}\quad\text{or}\quad x = 4 - 2\sqrt{5}
  6. 6

    As decimals: x ≈ 8.472136 and x ≈ -0.47213595.

  7. 7

    Check numerically: for x ≈ 8.472136 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 8x - 4 = 0\ \Rightarrow\ x = 4 + 2\sqrt{5},\ 4 - 2\sqrt{5} \]
\[ x^2 - 8x - 4 = 0 \]
\[ D = (-8)^2 - 4\cdot1\cdot(-4) = 64 + 16 = 80 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{80}}{2} \]
\[ x = 4 + 2\sqrt{5}\quad\text{or}\quad x = 4 - 2\sqrt{5} \]

The parabola y = x² - 8x - 4

xy
1-11
2-16
3-19
4-20
5-19
6-16
7-11

Questions

What are the solutions of x² - 8x - 4 = 0?

The solutions of x² - 8x - 4 = 0 are x = 4 + 2√5 or x = 4 - 2√5, that is x ≈ 8.472136 and x ≈ -0.47213595.

Does x² - 8x - 4 = 0 factor?

Not over the integers, because the discriminant 80 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 8x - 4?

(4, -20). 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 = -4.

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    Solve x² - 8x - 4 = 0: x = 4 + 2√5 or x = 4 - 2√5 (steps) | WordGPT