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Solve -3x² + 4x + 4 = 0

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

x² +x += 0
−3x2+4x+4=0⇒x=−23, 2-3x^2 + 4x + 4 = 0 \Rightarrow x = -\frac{2}{3},\ 2
Discriminant D = 64 · x ≈ -0.666667, 2

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

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Graph of y = -3x² + 4x + 4-3-2-11234-40-30-20-1010x = -0.66666667x = 2vertexy = -3x² + 4x + 4
Graph of y = -3x² + 4x + 4. The curve crosses the x-axis at x = -0.66666667 and x = 2; the vertex is at (0.66666667, 5.3333333).

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 = 4, c = 4.

    −3x2+4x+4=0-3x^2 + 4x + 4 = 0
  2. 2

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

    D=42−4⋅(−3)⋅4=16+48=64D = 4^2 - 4\cdot(-3)\cdot4 = 16 + 48 = 64
  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+2)(x−2)=0-\left(3x + 2\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+2=0⇒x=−23x−2=0⇒x=23x + 2 = 0 \Rightarrow x = -\frac{2}{3}\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−4±64−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{64}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−23orx=2x = -\frac{2}{3}\quad\text{or}\quad x = 2
  8. 8

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

LaTeX source of the formulas on this page
\[ -3x^2 + 4x + 4 = 0\ \Rightarrow\ x = -\frac{2}{3},\ 2 \]
\[ -3x^2 + 4x + 4 = 0 \]
\[ D = 4^2 - 4\cdot(-3)\cdot4 = 16 + 48 = 64 \]
\[ -\left(3x + 2\right)\left(x - 2\right) = 0 \]
\[ 3x + 2 = 0 \Rightarrow x = -\frac{2}{3}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{64}}{-6} \]
\[ x = -\frac{2}{3}\quad\text{or}\quad x = 2 \]

The parabola y = -3x² + 4x + 4

xy
-2.3333-21.666067
-1.3333-6.6662667
-0.33332.3335333
0.66675.3333333
1.66672.3331333
2.6667-6.6670667
3.6667-21.667267

Questions

What are the solutions of -3x² + 4x + 4 = 0?

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

Does -3x² + 4x + 4 = 0 factor?

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

What is the vertex of y = -3x² + 4x + 4?

(2/3, 16/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 4/3, product = c/a = -4/3.

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