What value of $g$ makes the equation true?
$(x+7)(x-4)=x^2+g x-28$
Answer (2)
The value of g that makes the equation true is 3, as derived from expanding the left side and comparing coefficients of the x terms. By expanding ( x + 7 ) ( x − 4 ) to get x 2 + 3 x − 28 , we find that the coefficient corresponding to g must equal 3. Therefore, g = 3 .
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Expand the left side of the equation: ( x + 7 ) ( x − 4 ) = x 2 + 3 x − 28 .
Compare the expanded form with the right side of the equation: x 2 + 3 x − 28 = x 2 + gx − 28 .
Equate the coefficients of the x term: 3 = g .
The value of g that makes the equation true is 3 .
Explanation
Analyze the problem We are given the equation ( x + 7 ) ( x − 4 ) = x 2 + gx − 28 and asked to find the value of g that makes the equation true. To do this, we need to expand the left side of the equation and then compare the coefficients of the x term with the right side of the equation.
Expand the left side First, let's expand the left side of the equation: ( x + 7 ) ( x − 4 ) = x ( x − 4 ) + 7 ( x − 4 ) = x 2 − 4 x + 7 x − 28 = x 2 + 3 x − 28
Compare coefficients Now, we compare the expanded form x 2 + 3 x − 28 with the right side of the equation x 2 + gx − 28 . We can see that the coefficient of the x term on the left side is 3, and the coefficient of the x term on the right side is g . Therefore, we must have g = 3 .
Final Answer Thus, the value of g that makes the equation true is 3.
Examples
Understanding how to expand and compare coefficients in polynomial equations is a fundamental skill in algebra. For example, engineers use polynomial equations to model the behavior of circuits and systems. By correctly expanding and comparing coefficients, they can determine the values of components that will make the system perform as desired. This ensures that electronic devices function correctly and efficiently.
