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Solve 4x² - 6x = 0

The solutions of 4x² - 6x = 0 are x = 3/2 or x = 0.

x² +x += 0
4x2−6x=0⇒x=32, 04x^2 - 6x = 0 \Rightarrow x = \frac{3}{2},\ 0
Discriminant D = 36 · x ≈ 1.5, 0

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

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Graph of y = 4x² - 6x-2-1123410203040x = 1.5x = 0vertexy = 4x² - 6x
Graph of y = 4x² - 6x. The curve crosses the x-axis at x = 1.5 and x = 0; the vertex is at (0.75, -2.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = -6, c = 0.

    4x2−6x=04x^2 - 6x = 0
  2. 2

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

    D=(−6)2−4⋅4⋅0=36+0=36D = (-6)^2 - 4\cdot4\cdot0 = 36 + 0 = 36
  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:

    2(2x−3)(x)=02\left(2x - 3\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:

    2x−3=0⇒x=32x=0⇒x=02x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=6±368x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{36}}{8}
  7. 7

    Simplify to get the solutions:

    x=32orx=0x = \frac{3}{2}\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = 3/2 back in: a·x² + b·x + c = 4·(9/4) + (-6)·(3/2) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 6x = 0\ \Rightarrow\ x = \frac{3}{2},\ 0 \]
\[ 4x^2 - 6x = 0 \]
\[ D = (-6)^2 - 4\cdot4\cdot0 = 36 + 0 = 36 \]
\[ 2\left(2x - 3\right)\left(x\right) = 0 \]
\[ 2x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{36}}{8} \]
\[ x = \frac{3}{2}\quad\text{or}\quad x = 0 \]

The parabola y = 4x² - 6x

xy
-2.2533.75
-1.2513.75
-0.251.75
0.75-2.25
1.751.75
2.7513.75
3.7533.75

Questions

What are the solutions of 4x² - 6x = 0?

The solutions of 4x² - 6x = 0 are x = 3/2 or x = 0.

Does 4x² - 6x = 0 factor?

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

What is the vertex of y = 4x² - 6x?

(3/4, -9/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3/2, product = c/a = 0.

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