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If $f(x)=5 x^{2}-3$ and $f(x+a)=5 x^{2}+30 x+42$, what is the value of $a$ ?

SAT · Math · previous-year question

  1. A.-30
  2. B.$\quad-3$
  3. C.3correct
  4. D.30

Answer

C. 3

Explanation

Choice C is correct. Substituting $x+a$ for $x$ in $f(x)=5 x^{2}-3$ yields $f(x+a)=5(x+a)^{2}-3$. Expanding the expression $5(x+a)^{2}$ by multiplication yields $5 x^{2}+10 a x+5 a^{2}$, and thus $f(x+a)=5 x^{2}+10 a x+5 a^{2}-3$. Setting the expression on the right-hand side of this equation equal to the given expression for $f(x+a)$ yields $5 x^{2}+30 x+42=5 x^{2}+10 a x+5 a^{2}-3$. Because this equality must be true for all values of $x$, the coefficients of each power of $x$ are equal. Setting the coefficients of $x$ equal to each other gives $10 a=30$. Dividing each side of this equation

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