Factor the trinomial completely.
[tex]7 x^2+28 x-84[/tex]
Answer (2)
Factor out the greatest common factor (GCF) 7: 7 x 2 + 28 x − 84 = 7 ( x 2 + 4 x − 12 ) .
Find two numbers that multiply to -12 and add to 4: these are 6 and -2.
Factor the quadratic expression: x 2 + 4 x − 12 = ( x + 6 ) ( x − 2 ) .
Write the complete factorization: 7 ( x + 6 ) ( x − 2 ) .
Explanation
Understanding the Problem We are given the trinomial 7 x 2 + 28 x − 84 and asked to factor it completely.
Factoring out the GCF First, we look for the greatest common factor (GCF) of the coefficients. The GCF of 7, 28, and -84 is 7. We factor out the 7 from the trinomial: 7 x 2 + 28 x − 84 = 7 ( x 2 + 4 x − 12 )
Factoring the Quadratic Now we need to factor the quadratic expression x 2 + 4 x − 12 . We are looking for two numbers that multiply to -12 and add to 4.
Finding the Numbers The two numbers that satisfy these conditions are 6 and -2, because 6 × ( − 2 ) = − 12 and 6 + ( − 2 ) = 4 . Therefore, we can write the quadratic as: x 2 + 4 x − 12 = ( x + 6 ) ( x − 2 )
Complete Factorization Finally, we substitute this back into the expression with the GCF factored out: 7 ( x 2 + 4 x − 12 ) = 7 ( x + 6 ) ( x − 2 ) Thus, the completely factored trinomial is 7 ( x + 6 ) ( x − 2 ) .
Examples
Factoring trinomials is a fundamental skill in algebra and is used in many real-world applications. For example, engineers use factoring to simplify complex equations when designing structures or analyzing systems. In business, factoring can help optimize costs and profits by breaking down revenue and expense models into simpler components. Understanding how to factor trinomials allows you to solve quadratic equations, which are used to model projectile motion, growth rates, and many other phenomena.
To factor the trinomial 7 x 2 + 28 x − 84 , first factor out the GCF, which is 7, yielding 7 ( x 2 + 4 x − 12 ) . Then, factor the quadratic x 2 + 4 x − 12 into ( x + 6 ) ( x − 2 ) . The complete factorization is 7 ( x + 6 ) ( x − 2 ) .
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