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Consider this expression:
[tex]$-3 x^2-24 x-36$[/tex]
What expression is equivalent to the given expression?
[tex]$v(x+\square v)(x+\square v)$[/tex]
Answer (2)
Factor out the constant term from the quadratic expression: − 3 x 2 − 24 x − 36 = − 3 ( x 2 + 8 x + 12 ) .
Factor the quadratic expression inside the parenthesis: x 2 + 8 x + 12 = ( x + 2 ) ( x + 6 ) .
Rewrite the expression as − 3 ( x + 2 ) ( x + 6 ) .
The equivalent expression is − 3 ( x + 2 ) ( x + 6 ) .
Explanation
Understanding the Problem We are given the quadratic expression − 3 x 2 − 24 x − 36 and we want to find an equivalent expression in the form v ( x + a ) ( x + b ) . Our goal is to factor the given expression and match it to the desired form.
Factoring out the Constant First, we factor out the common factor from the quadratic expression. In this case, the common factor is -3: − 3 x 2 − 24 x − 36 = − 3 ( x 2 + 8 x + 12 ) Now we need to factor the quadratic expression inside the parenthesis: x 2 + 8 x + 12 .
Factoring the Quadratic To factor the quadratic expression x 2 + 8 x + 12 , we look for two numbers that add up to 8 and multiply to 12. These numbers are 2 and 6. Therefore, we can write the quadratic expression as: x 2 + 8 x + 12 = ( x + 2 ) ( x + 6 ) So, the original expression becomes: − 3 x 2 − 24 x − 36 = − 3 ( x + 2 ) ( x + 6 )
Matching the Form Now we compare the factored expression − 3 ( x + 2 ) ( x + 6 ) with the desired form v ( x + a ) ( x + b ) . We can see that v = − 3 , a = 2 , and b = 6 . Therefore, the equivalent expression is: − 3 ( x + 2 ) ( x + 6 ) So the values to fill in the blanks are 2 and 6.
Final Answer The equivalent expression to the given expression is − 3 ( x + 2 ) ( x + 6 ) .
Examples
Factoring quadratic expressions is a fundamental skill in algebra and has numerous real-world applications. For example, engineers use factoring to analyze the stability of structures and to design efficient systems. Imagine you are designing a bridge, and you need to ensure that the forces acting on the bridge are balanced. By expressing these forces as a quadratic equation and factoring it, you can determine the critical points where the forces are most likely to cause stress. This allows you to reinforce those areas and ensure the bridge's structural integrity. Similarly, in physics, factoring is used to solve problems related to projectile motion and energy conservation.
The equivalent expression to − 3 x 2 − 24 x − 36 is − 3 ( x + 2 ) ( x + 6 ) , where the values to fill in are 2 and 6. This results from factoring out -3 and then factoring the quadratic expression. Thus, we can conclude that v = − 3 , a = 2 , and b = 6 .
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