Factor the following expression:
[tex]\begin{array}{r}
12 x^2-56 x+9 \\
(2 x-)(x-)
\end{array}[/tex]
Answer (1)
Find the roots of the quadratic equation using the quadratic formula.
Calculate the discriminant: b 2 − 4 a c = 2704 .
Determine the roots: x 1 = 2 9 and x 2 = 6 1 .
Write the factored form: ( 2 x − 9 ) ( 6 x − 1 ) .
Explanation
Understanding the Problem We are given a quadratic expression 12 x 2 − 56 x + 9 to factor. We are also given a partially factored form ( 2 x − 1 ) ( x − ) . Our goal is to find the complete factored form of the given quadratic expression.
Using the Quadratic Formula First, let's find the roots of the quadratic equation 12 x 2 − 56 x + 9 = 0 using the quadratic formula. The quadratic formula is given by x = 2 a − b \t p m \t s q r t b 2 − 4 a c , where a = 12 , b = − 56 , and c = 9 .
Calculating the Discriminant Calculate the discriminant: b 2 − 4 a c = ( − 56 ) 2 − 4 ( 12 ) ( 9 ) = 3136 − 432 = 2704 .
Finding the Roots Find the roots: x = 24 56 \t p m \t s q r t 2704 = 24 56 \t p m \t 52 . So the roots are x 1 = 24 56 + 52 = 24 108 = 2 9 and x 2 = 24 56 − 52 = 24 4 = 6 1 .
Writing the Factored Form Now we can write the factored form as 12 ( x − 2 9 ) ( x − 6 1 ) . Multiplying the constants through, we get ( 2 x − 9 ) ( 6 x − 1 ) .
Final Answer Therefore, the factored form of the given expression is ( 2 x − 9 ) ( 6 x − 1 ) .
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. Factoring helps simplify complex equations, making them easier to solve and understand. For instance, if you are designing a bridge, you might use a quadratic equation to model the load on the bridge. Factoring this equation can help you determine the maximum load the bridge can safely handle.
