Factor the following expression:
[tex]$\begin{array}{l}
7 x^2+15 x+2 \\
([?] x+[])(x+[])
\end{array}$[/tex]
Answer (2)
We are asked to factor the quadratic expression 7 x 2 + 15 x + 2 .
We look for a factored form of ( 7 x + b ) ( x + c ) such that ( 7 x + b ) ( x + c ) = 7 x 2 + 15 x + 2 .
We need to find b and c such that 7 c + b = 15 and b c = 2 .
By testing possible integer values, we find that b = 1 and c = 2 , so the factored form is ( 7 x + 1 ) ( x + 2 ) .
Explanation
Understanding the Problem We are given a quadratic expression 7 x 2 + 15 x + 2 and asked to factor it into the form ( a x + b ) ( x + c ) .
Setting up the Equation We need to find the values of a , b , and c such that ( a x + b ) ( x + c ) = 7 x 2 + 15 x + 2 .
Determining the Coefficient of x Since the coefficient of x 2 is 7, and one factor has x , the other factor must have 7 x . Thus, a = 7 .
Expanding the Factored Form So the factored form is ( 7 x + b ) ( x + c ) = 7 x 2 + ( 7 c + b ) x + b c = 7 x 2 + 15 x + 2 .
Finding b and c We need to find b and c such that 7 c + b = 15 and b c = 2 .
Possible Integer Values Since b c = 2 , the possible integer values for b and c are (1, 2) or (2, 1).
Testing Case 1 Case 1: If b = 1 and c = 2 , then 7 c + b = 7 ( 2 ) + 1 = 14 + 1 = 15 . This works.
Testing Case 2 Case 2: If b = 2 and c = 1 , then 7 c + b = 7 ( 1 ) + 2 = 7 + 2 = 9 = 15 . This does not work.
Determining b and c Therefore, b = 1 and c = 2 .
The Factored Form The factored form is ( 7 x + 1 ) ( x + 2 ) .
Verification Verify: ( 7 x + 1 ) ( x + 2 ) = 7 x 2 + 14 x + x + 2 = 7 x 2 + 15 x + 2 .
Final Answer Thus, the factored form 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, ensuring stability and optimal use of materials. Imagine designing a rectangular garden where the area is represented by the quadratic expression 7 x 2 + 15 x + 2 . By factoring this expression into ( 7 x + 1 ) ( x + 2 ) , you determine the dimensions of the garden, which helps in planning the layout and resource allocation. This skill is also crucial in physics for solving problems related to projectile motion and energy conservation.
The expression 7 x 2 + 15 x + 2 factors to ( 7 x + 1 ) ( x + 2 ) . We found suitable values for b and c that satisfy the conditions needed for factoring. Verification of the factorization confirms it as correct.
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