Factor: [tex]2 r^2+11 r+5[/tex]
Answer (2)
Find two numbers that multiply to a c and add up to b .
Rewrite the middle term using these two numbers.
Factor by grouping.
Factor out the common factor to get the factored form: ( 2 r + 1 ) ( r + 5 ) .
Explanation
Understanding the Problem We are asked to factor the quadratic expression 2 r 2 + 11 r + 5 . This is a quadratic in the form of a r 2 + b r + c , where a = 2 , b = 11 , and c = 5 .
Finding the Right Numbers We need to find two numbers that multiply to a c = 2 × 5 = 10 and add up to b = 11 .
Rewriting the Middle Term The two numbers are 10 and 1 since 10 × 1 = 10 and 10 + 1 = 11 . Now, we rewrite the middle term using these two numbers: 2 r 2 + 11 r + 5 = 2 r 2 + 10 r + 1 r + 5
Factoring by Grouping Next, we factor by grouping: ( 2 r 2 + 10 r ) + ( 1 r + 5 ) = 2 r ( r + 5 ) + 1 ( r + 5 )
Factoring out the Common Factor Finally, we factor out the common factor ( r + 5 ) : ( 2 r + 1 ) ( r + 5 ) Thus, the factored form of the quadratic expression is ( 2 r + 1 ) ( r + 5 ) .
Final Answer The factored form of the given quadratic expression is ( 2 r + 1 ) ( r + 5 ) .
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 calculate stress, while economists use it to model supply and demand curves. Imagine you are designing a rectangular garden with an area represented by the expression 2 r 2 + 11 r + 5 . By factoring this expression into ( 2 r + 1 ) ( r + 5 ) , you determine the dimensions of the garden, which helps you plan the layout and purchase the right amount of materials.
The expression 2 r 2 + 11 r + 5 can be factored into ( 2 r + 1 ) ( r + 5 ) by splitting the middle term and grouping. We find two numbers, 10 and 1, that multiply to 10 and add to 11 to help with this process. The final result shows the expression in its factored form.
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