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Solve -3x² - 4x + 4 = 0

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

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

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

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Graph of y = -3x² - 4x + 4-4-3-2-1123-40-30-20-1010x = -2x = 0.66666667vertexy = -3x² - 4x + 4
Graph of y = -3x² - 4x + 4. The curve crosses the x-axis at x = -2 and x = 0.66666667; 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=(−4)2−4⋅(−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:

    −(x+2)(3x−2)=0-\left(x + 2\right)\left(3x - 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+2=0⇒x=−23x−2=0⇒x=23x + 2 = 0 \Rightarrow x = -2\qquad 3x - 2 = 0 \Rightarrow x = \frac{2}{3}
  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=−2orx=23x = -2\quad\text{or}\quad x = \frac{2}{3}
  8. 8

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

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

The parabola y = -3x² - 4x + 4

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

Questions

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

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

Does -3x² - 4x + 4 = 0 factor?

Yes: -1(x+2)(3x-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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