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Solve -3x² + 4x = 0

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

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

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

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Graph of y = -3x² + 4x-2-1123-30-25-20-15-10-55x = 0x = 1.3333333vertexy = -3x² + 4x
Graph of y = -3x² + 4x. The curve crosses the x-axis at x = 0 and x = 1.3333333; 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=42−4⋅(−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:

    −(x)(3x−4)=0-\left(x\right)\left(3x - 4\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=0⇒x=03x−4=0⇒x=43x = 0 \Rightarrow x = 0\qquad 3x - 4 = 0 \Rightarrow x = \frac{4}{3}
  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=0orx=43x = 0\quad\text{or}\quad x = \frac{4}{3}
  8. 8

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

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

The parabola y = -3x² + 4x

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

Questions

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

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

Does -3x² + 4x = 0 factor?

Yes: -1(x)(3x-4) = 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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