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Solve x² - 8 = 0

The solutions of x² - 8 = 0 are x = 2√2 or x = -2√2, that is x ≈ 2.8284271 and x ≈ -2.8284271.

x² +x += 0
x2−8=0⇒x=22, −22x^2 - 8 = 0 \Rightarrow x = 2\sqrt{2},\ -2\sqrt{2}
Discriminant D = 32 · x ≈ 2.82843, -2.82843

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

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Graph of y = x² - 8-4-224-10-55101520x = 2.8284271x = -2.8284271vertexy = x² - 8
Graph of y = x² - 8. The curve crosses the x-axis at x = 2.8284271 and x = -2.8284271; the vertex is at (0, -8).

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 = 0, c = -8.

    x2−8=0x^2 - 8 = 0
  2. 2

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

    D=02−4⋅1⋅(−8)=0+32=32D = 0^2 - 4\cdot1\cdot(-8) = 0 + 32 = 32
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=0±322x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{32}}{2}
  5. 5

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

    x=22orx=−22x = 2\sqrt{2}\quad\text{or}\quad x = -2\sqrt{2}
  6. 6

    As decimals: x ≈ 2.8284271 and x ≈ -2.8284271.

  7. 7

    Check numerically: for x ≈ 2.8284271 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 8 = 0\ \Rightarrow\ x = 2\sqrt{2},\ -2\sqrt{2} \]
\[ x^2 - 8 = 0 \]
\[ D = 0^2 - 4\cdot1\cdot(-8) = 0 + 32 = 32 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{32}}{2} \]
\[ x = 2\sqrt{2}\quad\text{or}\quad x = -2\sqrt{2} \]

The parabola y = x² - 8

xy
-31
-2-4
-1-7
0-8
1-7
2-4
31

Questions

What are the solutions of x² - 8 = 0?

The solutions of x² - 8 = 0 are x = 2√2 or x = -2√2, that is x ≈ 2.8284271 and x ≈ -2.8284271.

Does x² - 8 = 0 factor?

Not over the integers, because the discriminant 32 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² - 8?

(0, -8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 0, product = c/a = -8.

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