Decide which is the correct factored form of [tex]$7 x^2-13 x-2$[/tex]
Answer (2)
The factored form of 7 x 2 − 13 x − 2 is ( 7 x + 1 ) ( x − 2 ) . This was derived by determining pairs of coefficients and testing combinations until the original expression was obtained. This process highlights the importance of factoring in algebra.
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We want to factor the quadratic expression 7 x 2 − 13 x − 2 .
We look for two binomials ( a x + b ) ( c x + d ) such that a c = 7 , a d + b c = − 13 , and b d = − 2 .
By trial and error, we find that ( 7 x + 1 ) ( x − 2 ) satisfies these conditions.
Therefore, the factored form is ( 7 x + 1 ) ( x − 2 ) .
Explanation
Understanding the Problem We are given the quadratic expression 7 x 2 − 13 x − 2 . Our goal is to find its factored form. We are looking for two binomials that multiply to give this quadratic.
Setting up the Factoring We want to factor the quadratic expression 7 x 2 − 13 x − 2 . This means we want to find two binomials ( a x + b ) and ( c x + d ) such that ( a x + b ) ( c x + d ) = 7 x 2 − 13 x − 2 .
Matching Coefficients When we expand ( a x + b ) ( c x + d ) , we get a c x 2 + ( a d + b c ) x + b d . Comparing this to 7 x 2 − 13 x − 2 , we need to find a , b , c , d such that a c = 7 , a d + b c = − 13 , and b d = − 2 .
Finding Possible Values Since 7 is a prime number, the possible values for a and c are 7 and 1 (or -7 and -1). The possible values for b and d are 1 and -2, or -1 and 2. We can test different combinations of these values to find the correct factorization.
Testing a Combination Let's try ( 7 x + 1 ) ( x − 2 ) . Expanding this, we get 7 x 2 − 14 x + x − 2 = 7 x 2 − 13 x − 2 . This matches the given quadratic expression.
Final Answer Therefore, the correct factored form of 7 x 2 − 13 x − 2 is ( 7 x + 1 ) ( x − 2 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to design structures and solve problems related to stress and strain. In business, factoring can be used to analyze revenue and cost functions to determine break-even points. Imagine you are designing a bridge and need to calculate the load it can bear. The equation describing the load might be a quadratic, and factoring it helps you find the critical points where the load is maximized or minimized, ensuring the bridge's safety.
