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Solve -3x² - 8x - 4 = 0

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

x² +x += 0
−3x2−8x−4=0⇒x=−2, −23-3x^2 - 8x - 4 = 0 \Rightarrow x = -2,\ -\frac{2}{3}
Discriminant D = 16 · 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 - 8x - 4-4-3-2-11-30-25-20-15-10-55x = -2x = -0.66666667vertexy = -3x² - 8x - 4
Graph of y = -3x² - 8x - 4. The curve crosses the x-axis at x = -2 and x = -0.66666667; the vertex is at (-1.3333333, 1.3333333).

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 = -8, c = -4.

    −3x2−8x−4=0-3x^2 - 8x - 4 = 0
  2. 2

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

    D=(−8)2−4⋅(−3)⋅(−4)=64−48=16D = (-8)^2 - 4\cdot(-3)\cdot(-4) = 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:

    −(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=8±16−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{16}}{-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) + (-8)·(-2) + (-4) = 0 ✓

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

The parabola y = -3x² - 8x - 4

xy
-4.3333-25.666067
-3.3333-10.666267
-2.3333-1.6664667
-1.33331.3333333
-0.3333-1.6668667
0.6667-10.667067
1.6667-25.667267

Questions

What are the solutions of -3x² - 8x - 4 = 0?

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

Does -3x² - 8x - 4 = 0 factor?

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

What is the vertex of y = -3x² - 8x - 4?

(-4/3, 4/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -8/3, product = c/a = 4/3.

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