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Solve -3x² - 4x = 0

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

x² +x += 0
−3x2−4x=0⇒x=−43, 0-3x^2 - 4x = 0 \Rightarrow x = -\frac{4}{3},\ 0
Discriminant D = 16 · x ≈ -1.33333, 0

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

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Graph of y = -3x² - 4x-3-2-112-30-25-20-15-10-55x = -1.3333333x = 0vertexy = -3x² - 4x
Graph of y = -3x² - 4x. The curve crosses the x-axis at x = -1.3333333 and x = 0; the vertex is at (-0.66666667, 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 = -4, c = 0.

    −3x2−4x=0-3x^2 - 4x = 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)⋅0=16+0=16D = (-4)^2 - 4\cdot(-3)\cdot0 = 16 + 0 = 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:

    −(3x+4)(x)=0-\left(3x + 4\right)\left(x\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=0⇒x=03x + 4 = 0 \Rightarrow x = -\frac{4}{3}\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=4±16−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{16}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−43orx=0x = -\frac{4}{3}\quad\text{or}\quad x = 0
  8. 8

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

LaTeX source of the formulas on this page
\[ -3x^2 - 4x = 0\ \Rightarrow\ x = -\frac{4}{3},\ 0 \]
\[ -3x^2 - 4x = 0 \]
\[ D = (-4)^2 - 4\cdot(-3)\cdot0 = 16 + 0 = 16 \]
\[ -\left(3x + 4\right)\left(x\right) = 0 \]
\[ 3x + 4 = 0 \Rightarrow x = -\frac{4}{3}\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{16}}{-6} \]
\[ x = -\frac{4}{3}\quad\text{or}\quad x = 0 \]

The parabola y = -3x² - 4x

xy
-3.6667-25.667267
-2.6667-10.667067
-1.6667-1.6668667
-0.66671.3333333
0.3333-1.6664667
1.3333-10.666267
2.3333-25.666067

Questions

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

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

Does -3x² - 4x = 0 factor?

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

What is the vertex of y = -3x² - 4x?

(-2/3, 4/3). 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 = 0.

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