Question

Factor the expression. $$-48 x^{2}+216 x-243$$

Short answer

The factor of the expression \(-48x^{2}+216x-243\) is \(-3(4x-9)^2\)

Step-by-step solution

Identify and Factor out the Greatest Common Factor (GCF)

The GCF among the three terms \(-48x^{2}\), \(216x\), and \(-243\) is \(-3\). To factor it out, divide each term by \(-3\), which yields \[16x^2-72x+81\]. Thus, the initial expression becomes \(-3(16x^2-72x+81)\].

Factor the quadratic part

The quadratic \(16x^2-72x+81\) can be factored into \((4x-9)^2\), a perfect square. To see this, recognize that 16, the coefficient of \(x^2\), is \(4^2\), and 81 is \(9^2\), and \(-2*4*9\) equals \(-72\), the coefficient of x. Hence, we can rewrite the quadratic as \((4x-9)^2\).

Combine the factors

Combine the factored GCF from Step 1 with the factored quadratic from Step 2. The fully factored expression is: \[-3(4x-9)^2\]