Solve 54a-48a^2

$$54 a - 48 a ^ { 2 }$$

Factor

$$6a\left(9-8a\right)$$

Solution Steps

Factor out 6.

$$6\left(9a-8a^{2}\right)$$

Consider 9a-8a^{2}. Factor out a.

$$a\left(9-8a\right)$$

Rewrite the complete factored expression.

$$6a\left(-8a+9\right)$$

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.

$$-48a^{2}+54a=0$$

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.

$$a=\frac{-54±\sqrt{54^{2}}}{2\left(-48\right)}$$

Take the square root of 54^{2}.

$$a=\frac{-54±54}{2\left(-48\right)}$$

Multiply 2 times -48.

$$a=\frac{-54±54}{-96}$$

Now solve the equation a=\frac{-54±54}{-96} when ± is plus. Add -54 to 54.

$$a=\frac{0}{-96}$$

Divide 0 by -96.

$$a=0$$

Now solve the equation a=\frac{-54±54}{-96} when ± is minus. Subtract 54 from -54.

$$a=-\frac{108}{-96}$$

Reduce the fraction \frac{-108}{-96} to lowest terms by extracting and canceling out 12.

$$a=\frac{9}{8}$$

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{9}{8} for x_{2}.

$$-48a^{2}+54a=-48a\left(a-\frac{9}{8}\right)$$

Subtract \frac{9}{8} from a by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-48a^{2}+54a=-48a\times \frac{-8a+9}{-8}$$

Cancel out 8, the greatest common factor in -48 and -8.

$$-48a^{2}+54a=6a\left(-8a+9\right)$$

Evaluate

$$6a\left(9-8a\right)$$