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Solve -3x² + 2x + 8 = 0

The solutions of -3x² + 2x + 8 = 0 are x = -4/3 or x = 2.

x² +x += 0
−3x2+2x+8=0⇒x=−43, 2-3x^2 + 2x + 8 = 0 \Rightarrow x = -\frac{4}{3},\ 2
Discriminant D = 100 · x ≈ -1.33333, 2

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

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Graph of y = -3x² + 2x + 8-3-2-11234-40-30-20-1010x = -1.3333333x = 2vertexy = -3x² + 2x + 8
Graph of y = -3x² + 2x + 8. The curve crosses the x-axis at x = -1.3333333 and x = 2; the vertex is at (0.33333333, 8.3333333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = 2, c = 8.

    −3x2+2x+8=0-3x^2 + 2x + 8 = 0
  2. 2

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

    D=22−4⋅(−3)⋅8=4+96=100D = 2^2 - 4\cdot(-3)\cdot8 = 4 + 96 = 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:

    −(3x+4)(x−2)=0-\left(3x + 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:

    3x+4=0⇒x=−43x−2=0⇒x=23x + 4 = 0 \Rightarrow x = -\frac{4}{3}\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

    x=−43orx=2x = -\frac{4}{3}\quad\text{or}\quad x = 2
  8. 8

    Check by putting x = -4/3 back in: a·x² + b·x + c = -3·(16/9) + 2·(-4/3) + 8 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 2x + 8 = 0\ \Rightarrow\ x = -\frac{4}{3},\ 2 \]
\[ -3x^2 + 2x + 8 = 0 \]
\[ D = 2^2 - 4\cdot(-3)\cdot8 = 4 + 96 = 100 \]
\[ -\left(3x + 4\right)\left(x - 2\right) = 0 \]
\[ 3x + 4 = 0 \Rightarrow x = -\frac{4}{3}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm \sqrt{100}}{-6} \]
\[ x = -\frac{4}{3}\quad\text{or}\quad x = 2 \]

The parabola y = -3x² + 2x + 8

xy
-2.6667-18.667267
-1.6667-3.6670667
-0.66675.3331333
0.33338.3333333
1.33335.3335333
2.3333-3.6662667
3.3333-18.666067

Questions

What are the solutions of -3x² + 2x + 8 = 0?

The solutions of -3x² + 2x + 8 = 0 are x = -4/3 or x = 2.

Does -3x² + 2x + 8 = 0 factor?

Yes: -1(3x+4)(x-2) = 0.

What is the vertex of y = -3x² + 2x + 8?

(1/3, 25/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 2/3, product = c/a = -8/3.

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