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Solve x² - 2x - 8 = 0

The solutions of x² - 2x - 8 = 0 are x = 4 or x = -2.

x² +x += 0
x2−2x−8=0⇒x=4, −2x^2 - 2x - 8 = 0 \Rightarrow x = 4,\ -2
Discriminant D = 36 · x ≈ 4, -2

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

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Graph of y = x² - 2x - 8-4-2246-10-55101520x = 4x = -2vertexy = x² - 2x - 8
Graph of y = x² - 2x - 8. The curve crosses the x-axis at x = 4 and x = -2; the vertex is at (1, -9).

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 = -2, c = -8.

    x2−2x−8=0x^2 - 2x - 8 = 0
  2. 2

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

    D=(−2)2−4⋅1⋅(−8)=4+32=36D = (-2)^2 - 4\cdot1\cdot(-8) = 4 + 32 = 36
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x−4)(x+2)=0\left(x - 4\right)\left(x + 2\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x−4=0⇒x=4x+2=0⇒x=−2x - 4 = 0 \Rightarrow x = 4\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=2±362x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{36}}{2}
  7. 7

    Simplify to get the solutions:

    x=4orx=−2x = 4\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = 4 back in: a·x² + b·x + c = 1·(16) + (-2)·(4) + (-8) = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - 2x - 8 = 0\ \Rightarrow\ x = 4,\ -2 \]
\[ x^2 - 2x - 8 = 0 \]
\[ D = (-2)^2 - 4\cdot1\cdot(-8) = 4 + 32 = 36 \]
\[ \left(x - 4\right)\left(x + 2\right) = 0 \]
\[ x - 4 = 0 \Rightarrow x = 4\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{36}}{2} \]
\[ x = 4\quad\text{or}\quad x = -2 \]

The parabola y = x² - 2x - 8

xy
-20
-1-5
0-8
1-9
2-8
3-5
40

Questions

What are the solutions of x² - 2x - 8 = 0?

The solutions of x² - 2x - 8 = 0 are x = 4 or x = -2.

Does x² - 2x - 8 = 0 factor?

Yes: (x-4)(x+2) = 0.

What is the vertex of y = x² - 2x - 8?

(1, -9). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 2, product = c/a = -8.

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