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Solve 4x² - 3x = 0

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

x² +x += 0
4x2−3x=0⇒x=34, 04x^2 - 3x = 0 \Rightarrow x = \frac{3}{4},\ 0
Discriminant D = 9 · x ≈ 0.75, 0

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

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Graph of y = 4x² - 3x-2-11235101520253035x = 0.75x = 0vertexy = 4x² - 3x
Graph of y = 4x² - 3x. The curve crosses the x-axis at x = 0.75 and x = 0; the vertex is at (0.375, -0.5625).

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 = -3, c = 0.

    4x2−3x=04x^2 - 3x = 0
  2. 2

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

    D=(−3)2−4⋅4⋅0=9+0=9D = (-3)^2 - 4\cdot4\cdot0 = 9 + 0 = 9
  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:

    (4x−3)(x)=0\left(4x - 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:

    4x−3=0⇒x=34x=0⇒x=04x - 3 = 0 \Rightarrow x = \frac{3}{4}\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±98x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{9}}{8}
  7. 7

    Simplify to get the solutions:

    x=34orx=0x = \frac{3}{4}\quad\text{or}\quad x = 0
  8. 8

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

LaTeX source of the formulas on this page
\[ 4x^2 - 3x = 0\ \Rightarrow\ x = \frac{3}{4},\ 0 \]
\[ 4x^2 - 3x = 0 \]
\[ D = (-3)^2 - 4\cdot4\cdot0 = 9 + 0 = 9 \]
\[ \left(4x - 3\right)\left(x\right) = 0 \]
\[ 4x - 3 = 0 \Rightarrow x = \frac{3}{4}\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{9}}{8} \]
\[ x = \frac{3}{4}\quad\text{or}\quad x = 0 \]

The parabola y = 4x² - 3x

xy
-2.62535.4375
-1.62515.4375
-0.6253.4375
0.375-0.5625
1.3753.4375
2.37515.4375
3.37535.4375

Questions

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

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

Does 4x² - 3x = 0 factor?

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

What is the vertex of y = 4x² - 3x?

(3/8, -9/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3/4, product = c/a = 0.

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