Question

If $f\left(x\right)=$ $x^2+x$ for all $x$ and if $f(a-1)=$ $-\frac{1}{4}$, find the value of $a$.

• A. $-\frac{1}{2}$
• B. $-\frac{1}{4}$
• C. $\frac{1}{4}$
• D. $\frac{1}{2}$
• E. $\frac{3}{4}$

Answer 1

The correct option is D $\frac{1}{2}$

Given $f\left(x\right)=x^2+x$ and $f(a-1)=-1/4$

When $x=a-1,$ $x^2=a^2-2a+1$

$\Rightarrow f(a-1)={(a-1)}^2+(a-1)=a^2-2a+1+a-1=a^2-a$

$\Rightarrow f(a-1)=\frac{-1}{4}$ is given.

Then $a^2-a=\frac{-1}{4}$

$\Rightarrow 4a^2-4a+1=0$

$\Rightarrow (2a-1{)}^2=0$

$\Rightarrow a=\frac{1}{2}$