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Solve -2x² + 8x - 6 = 0

The solutions of -2x² + 8x - 6 = 0 are x = 1 or x = 3.

x² +x += 0
−2x2+8x−6=0⇒x=1, 3-2x^2 + 8x - 6 = 0 \Rightarrow x = 1,\ 3
Discriminant D = 16 · x ≈ 1, 3

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

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Graph of y = -2x² + 8x - 6-112345-20-15-10-55x = 1x = 3vertexy = -2x² + 8x - 6
Graph of y = -2x² + 8x - 6. The curve crosses the x-axis at x = 1 and x = 3; the vertex is at (2, 2).

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 = 8, c = -6.

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

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

    D=82−4⋅(−2)⋅(−6)=64−48=16D = 8^2 - 4\cdot(-2)\cdot(-6) = 64 - 48 = 16
  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−1)(x−3)=0-2\left(x - 1\right)\left(x - 3\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−1=0⇒x=1x−3=0⇒x=3x - 1 = 0 \Rightarrow x = 1\qquad x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±16−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{16}}{-4}
  7. 7

    Simplify to get the solutions:

    x=1orx=3x = 1\quad\text{or}\quad x = 3
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = -2·(1) + 8·(1) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 8x - 6 = 0\ \Rightarrow\ x = 1,\ 3 \]
\[ -2x^2 + 8x - 6 = 0 \]
\[ D = 8^2 - 4\cdot(-2)\cdot(-6) = 64 - 48 = 16 \]
\[ -2\left(x - 1\right)\left(x - 3\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{16}}{-4} \]
\[ x = 1\quad\text{or}\quad x = 3 \]

The parabola y = -2x² + 8x - 6

xy
-1-16
0-6
10
22
30
4-6
5-16

Questions

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

The solutions of -2x² + 8x - 6 = 0 are x = 1 or x = 3.

Does -2x² + 8x - 6 = 0 factor?

Yes: -2(x-1)(x-3) = 0.

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

(2, 2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 4, product = c/a = 3.

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