Factor the following expression.
[tex]\begin{array}{r}
4 x^2-9 x-9 \\
(x-\square)(\square x+\square)
\end{array}[/tex]
Answer (1)
We are given the quadratic expression 4 x 2 − 9 x − 9 and asked to factor it.
We expand the factored form ( x − a ) ( b x + c ) and equate coefficients to get b = 4 and c = 4 a − 9 .
We solve the quadratic equation 4 a 2 − 9 a − 9 = 0 to find a = 3 or a = − 4 3 .
We find the corresponding value of c and verify the factorization, which gives us the final answer: ( x − 3 ) ( 4 x + 3 ) .
Explanation
Understanding the Problem We are given the quadratic expression 4 x 2 − 9 x − 9 and asked to factor it into the form ( x − a ) ( b x + c ) . Our goal is to find the values of a , b , c such that ( x − a ) ( b x + c ) = 4 x 2 − 9 x − 9 .
Setting up Equations Expanding ( x − a ) ( b x + c ) gives b x 2 + ( c − ab ) x − a c . We need to equate the coefficients of the corresponding terms in b x 2 + ( c − ab ) x − a c and 4 x 2 − 9 x − 9 . This gives us the equations b = 4 , c − ab = − 9 , and − a c = − 9 .
Substitution Substituting b = 4 into c − ab = − 9 gives c − 4 a = − 9 , so c = 4 a − 9 . Substituting c = 4 a − 9 into − a c = − 9 gives − a ( 4 a − 9 ) = − 9 , which simplifies to 4 a 2 − 9 a − 9 = 0 .
Solving for a We need to solve the quadratic equation 4 a 2 − 9 a − 9 = 0 for a . We can use the quadratic formula or factoring. Let's try to factor it. We are looking for two numbers that multiply to 4 \t \t × − 9 = − 36 and add up to − 9 . These numbers are − 12 and 3 . So we can rewrite the middle term as − 12 a + 3 a . Thus, 4 a 2 − 12 a + 3 a − 9 = 0 . Factoring by grouping, we get 4 a ( a − 3 ) + 3 ( a − 3 ) = 0 , which gives ( 4 a + 3 ) ( a − 3 ) = 0 . Therefore, a = 3 or a = − 4 3 = − 0.75 .
Solving for c If a = 3 , then c = 4 ( 3 ) − 9 = 12 − 9 = 3 . So one possible factorization is ( x − 3 ) ( 4 x + 3 ) . If a = − 0.75 , then c = 4 ( − 0.75 ) − 9 = − 3 − 9 = − 12 . So another possible factorization is ( x + 0.75 ) ( 4 x − 12 ) which is the same as ( x + 4 3 ) ( 4 x − 12 ) . Multiplying this out gives 4 x 2 − 12 x + 3 x − 9 = 4 x 2 − 9 x − 9 .
Verification We can check the factorization ( x − 3 ) ( 4 x + 3 ) = 4 x 2 + 3 x − 12 x − 9 = 4 x 2 − 9 x − 9 . This matches the original expression. Thus, the factorization is ( x − 3 ) ( 4 x + 3 ) .
Final Answer Therefore, the factored form of the expression 4 x 2 − 9 x − 9 is ( x − 3 ) ( 4 x + 3 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to analyze the stability of structures, economists use it to model supply and demand curves, and computer scientists use it to design efficient algorithms. Suppose you want to build a rectangular garden with an area represented by the expression 4 x 2 − 9 x − 9 . By factoring this expression into ( x − 3 ) ( 4 x + 3 ) , you can determine the possible dimensions of the garden in terms of x . This allows you to plan the layout of your garden based on the available space and desired area.
