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Solve 2x² - 6x - 8 = 0

The solutions of 2x² - 6x - 8 = 0 are x = 4 or x = -1.

x² +x += 0
2x2−6x−8=0⇒x=4, −12x^2 - 6x - 8 = 0 \Rightarrow x = 4,\ -1
Discriminant D = 100 · x ≈ 4, -1

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

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Graph of y = 2x² - 6x - 8-3-2-1123456-1010203040x = 4x = -1vertexy = 2x² - 6x - 8
Graph of y = 2x² - 6x - 8. The curve crosses the x-axis at x = 4 and x = -1; the vertex is at (1.5, -12.5).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = -6, c = -8.

    2x2−6x−8=02x^2 - 6x - 8 = 0
  2. 2

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

    D=(−6)2−4⋅2⋅(−8)=36+64=100D = (-6)^2 - 4\cdot2\cdot(-8) = 36 + 64 = 100
  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:

    2(x−4)(x+1)=02\left(x - 4\right)\left(x + 1\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+1=0⇒x=−1x - 4 = 0 \Rightarrow x = 4\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=6±1004x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{100}}{4}
  7. 7

    Simplify to get the solutions:

    x=4orx=−1x = 4\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 4 back in: a·x² + b·x + c = 2·(16) + (-6)·(4) + (-8) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 6x - 8 = 0\ \Rightarrow\ x = 4,\ -1 \]
\[ 2x^2 - 6x - 8 = 0 \]
\[ D = (-6)^2 - 4\cdot2\cdot(-8) = 36 + 64 = 100 \]
\[ 2\left(x - 4\right)\left(x + 1\right) = 0 \]
\[ x - 4 = 0 \Rightarrow x = 4\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{100}}{4} \]
\[ x = 4\quad\text{or}\quad x = -1 \]

The parabola y = 2x² - 6x - 8

xy
-1.55.5
-0.5-4.5
0.5-10.5
1.5-12.5
2.5-10.5
3.5-4.5
4.55.5

Questions

What are the solutions of 2x² - 6x - 8 = 0?

The solutions of 2x² - 6x - 8 = 0 are x = 4 or x = -1.

Does 2x² - 6x - 8 = 0 factor?

Yes: 2(x-4)(x+1) = 0.

What is the vertex of y = 2x² - 6x - 8?

(3/2, -25/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = -4.

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