Factor the following expression.
[tex]
\begin{array}{r}
4 x^2-9 x-9 \\
(x-1)(x+)
\end{array}
[/tex]
Answer (1)
Find two numbers that multiply to 4 × − 9 = − 36 and add up to − 9 , which are − 12 and 3 .
Rewrite the middle term: 4 x 2 − 12 x + 3 x − 9 .
Factor by grouping: ( 4 x 2 − 12 x ) + ( 3 x − 9 ) = 4 x ( x − 3 ) + 3 ( x − 3 ) .
Factor out the common binomial: ( 4 x + 3 ) ( x − 3 ) . The factored form is ( 4 x + 3 ) ( x − 3 ) .
Explanation
Understanding the Problem We are given the quadratic expression 4 x 2 − 9 x − 9 and asked to factor it. Factoring involves rewriting the expression as a product of simpler expressions. In this case, we want to find two binomials ( a x + b ) and ( c x + d ) such that their product equals the given quadratic expression.
Finding the Right Numbers To factor the quadratic expression 4 x 2 − 9 x − 9 , we look for two numbers that multiply to 4 × − 9 = − 36 and add up to − 9 .
Identifying the Numbers The two numbers that satisfy these conditions are − 12 and 3 , since − 12 × 3 = − 36 and − 12 + 3 = − 9 .
Rewriting the Middle Term Now, we rewrite the middle term of the quadratic expression using these two numbers: 4 x 2 − 12 x + 3 x − 9 .
Factoring by Grouping Next, we factor by grouping. We group the first two terms and the last two terms: ( 4 x 2 − 12 x ) + ( 3 x − 9 ) .
Factoring out Common Factors We factor out the greatest common factor from each group: 4 x ( x − 3 ) + 3 ( x − 3 ) .
Factoring out the Common Binomial Now, we factor out the common binomial factor ( x − 3 ) : ( 4 x + 3 ) ( x − 3 ) .
Final Factorization Therefore, the factored form of the expression 4 x 2 − 9 x − 9 is ( 4 x + 3 ) ( x − 3 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has many real-world applications. For example, engineers use factoring to design structures and predict their behavior under different loads. Imagine designing a bridge; the load on the bridge can be modeled as a quadratic expression, and factoring helps determine the critical points where the structure might be most stressed. Similarly, in physics, factoring can help solve problems involving projectile motion, where the height of an object is described by a quadratic equation. By factoring, we can find when the object hits the ground or reaches its maximum height. This skill is also crucial in economics for optimizing costs and revenues, where quadratic functions often model these relationships.
