Question

How do you factor $9{x^2} - 24x + 16$?

Answer 1

Hint: In this algebraic problem, we have given some quadratic expression. Here we are asked to find the factors of the given quadratic expression. For this we have found the factors of the coefficients of ${x^2}$ term and the constant term and whose products add to the coefficient of $x$. By this way we can find the factors.

Complete step-by-step solution:
The given quadratic expression is $9{x^2} - 24x + 16$.
And the given quadratic expression is a perfect square trinomial. Because the given expression is of the form ${a^2} - 2ab + {b^2} = {(a - b)^2} - - - - - (1)$
Now compare the given quadratic expression and the equation (1), we get
$a = 3x$,$b = 4$ and $9{x^2} - 24x + 16 = {(3x)^2} - (2 \times (3x) \times 4) + {4^2}$
$ = {(3x - 4)^2}$

Therefore, the required answer is ${(3x - 4)^2}$

Additional Information: A quadratic expression is a polynomial function of degree $2$. A quadratic function is of the form $f(x) = a{x^2} + bx + c$ where a, b and c are constants. The term $a{x^2}$ is called the quadratic term, the term $bx$ is called the linear term, and the term c is called the constant term. The easiest method of solving a quadratic equation is factoring. Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation. If a quadratic equation can be factored, it is written as a product of linear terms.

Note: There are four methods of factoring. The following methods will be used to factor an expression that is factoring out the greatest common factor, the sum – product pattern, the grouping method, the perfect square trinomial pattern and the difference of squares pattern. Let us solve the given expression by using these ways,
Use the sum product pattern $9{x^2} - 24x + 16$
$ \Rightarrow 9{x^2} - 12x - 12x + 16$
Common factor from the two pairs
$ \Rightarrow 9{x^2} - 12x - 12x + 16$
$ \Rightarrow 3x(3x - 4) - 4(3x - 4)$
Rewrite in factored form
$ \Rightarrow 3x(3x - 4) - 4(3x - 4)$
$ \Rightarrow \left( {3x - 4} \right)\left( {3x - 4} \right)$
Combine to a square
$ \Rightarrow \left( {3x - 4} \right)\left( {3x - 4} \right)$
$ \Rightarrow {(3x - 4)^2}$
This is the required answer.