Solve 54x-36x^2

$$54 x - 36 x ^ { 2 }$$

Factor

$$18x\left(3-2x\right)$$

Solution Steps

Factor out 18.

$$18\left(3x-2x^{2}\right)$$

Consider 3x-2x^{2}. Factor out x.

$$x\left(3-2x\right)$$

Rewrite the complete factored expression.

$$18x\left(-2x+3\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-36x^{2}+54x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-54±\sqrt{54^{2}}}{2\left(-36\right)}$$

Take the square root of 54^{2}.

$$x=\frac{-54±54}{2\left(-36\right)}$$

Multiply 2 times -36.

$$x=\frac{-54±54}{-72}$$

Now solve the equation x=\frac{-54±54}{-72} when ± is plus. Add -54 to 54.

$$x=\frac{0}{-72}$$

Divide 0 by -72.

$$x=0$$

Now solve the equation x=\frac{-54±54}{-72} when ± is minus. Subtract 54 from -54.

$$x=-\frac{108}{-72}$$

Reduce the fraction \frac{-108}{-72} to lowest terms by extracting and canceling out 36.

$$x=\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and \frac{3}{2} for x_{2}.

$$-36x^{2}+54x=-36x\left(x-\frac{3}{2}\right)$$

Subtract \frac{3}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-36x^{2}+54x=-36x\times \frac{-2x+3}{-2}$$

Cancel out 2, the greatest common factor in -36 and -2.

$$-36x^{2}+54x=18x\left(-2x+3\right)$$

Evaluate

$$18x\left(3-2x\right)$$