Factor the quadratic $4 x^2-7 x-2$
List ONLY ONE of the factors with parentheses and no spaces. Example: $(x+1)$
Answer (1)
Find two numbers that multiply to 4 ⋅ ( − 2 ) = − 8 and add up to − 7 , which are − 8 and 1 .
Rewrite the middle term: 4 x 2 − 8 x + x − 2 .
Factor by grouping: 4 x ( x − 2 ) + 1 ( x − 2 ) .
Factor out the common factor: ( 4 x + 1 ) ( x − 2 ) . The answer is ( 4 x + 1 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression 4 x 2 − 7 x − 2 . Factoring a quadratic means rewriting it as a product of two binomials.
Finding the Right Numbers To factor the quadratic 4 x 2 − 7 x − 2 , we look for two numbers that multiply to 4 × ( − 2 ) = − 8 and add up to − 7 . These numbers are − 8 and 1 .
Rewriting the Middle Term We rewrite the middle term using these two numbers: 4 x 2 − 8 x + x − 2 .
Factoring by Grouping Now, we factor by grouping: 4 x ( x − 2 ) + 1 ( x − 2 ) .
Factoring out the Common Factor We factor out the common factor ( x − 2 ) : ( 4 x + 1 ) ( x − 2 ) .
Listing One Factor Therefore, the factored form of the quadratic 4 x 2 − 7 x − 2 is ( 4 x + 1 ) ( x − 2 ) . We are asked to list only one of the factors.
Examples
Factoring quadratics is a fundamental skill in algebra with applications in various fields. 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 in algorithm design. Consider a simple scenario where you want to find the dimensions of a rectangular garden with an area represented by the quadratic expression 4 x 2 − 7 x − 2 . By factoring this expression, you can determine the possible lengths and widths of the garden in terms of x .
