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Solve 4x² - 9x = 0

The solutions of 4x² - 9x = 0 are x = 9/4 or x = 0.

x² +x += 0
4x2−9x=0⇒x=94, 04x^2 - 9x = 0 \Rightarrow x = \frac{9}{4},\ 0
Discriminant D = 81 · x ≈ 2.25, 0

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

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Graph of y = 4x² - 9x-2-11234-101020304050x = 2.25x = 0vertexy = 4x² - 9x
Graph of y = 4x² - 9x. The curve crosses the x-axis at x = 2.25 and x = 0; the vertex is at (1.125, -5.0625).

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 = -9, c = 0.

    4x2−9x=04x^2 - 9x = 0
  2. 2

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

    D=(−9)2−4⋅4⋅0=81+0=81D = (-9)^2 - 4\cdot4\cdot0 = 81 + 0 = 81
  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−9)(x)=0\left(4x - 9\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−9=0⇒x=94x=0⇒x=04x - 9 = 0 \Rightarrow x = \frac{9}{4}\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=9±818x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{81}}{8}
  7. 7

    Simplify to get the solutions:

    x=94orx=0x = \frac{9}{4}\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = 9/4 back in: a·x² + b·x + c = 4·(81/16) + (-9)·(9/4) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 9x = 0\ \Rightarrow\ x = \frac{9}{4},\ 0 \]
\[ 4x^2 - 9x = 0 \]
\[ D = (-9)^2 - 4\cdot4\cdot0 = 81 + 0 = 81 \]
\[ \left(4x - 9\right)\left(x\right) = 0 \]
\[ 4x - 9 = 0 \Rightarrow x = \frac{9}{4}\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{81}}{8} \]
\[ x = \frac{9}{4}\quad\text{or}\quad x = 0 \]

The parabola y = 4x² - 9x

xy
-1.87530.9375
-0.87510.9375
0.125-1.0625
1.125-5.0625
2.125-1.0625
3.12510.9375
4.12530.9375

Questions

What are the solutions of 4x² - 9x = 0?

The solutions of 4x² - 9x = 0 are x = 9/4 or x = 0.

Does 4x² - 9x = 0 factor?

Yes: (4x-9)(x) = 0.

What is the vertex of y = 4x² - 9x?

(9/8, -81/16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 9/4, product = c/a = 0.

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