Question

If $\frac{a + 1}{a} = k$, what is the value of $\frac{a^2 - 1}{a^2}$?

• A. $k-2k^2$
• B. $k^2+2$
• C. $2k-k^2$
• D. $k+2$

Answer 1

Answer:

$2k-k^2$

The value of the expression is $2k - k^2$ (or $k(2 - k)$).

1: Simplify the given equation

Separate the fraction into two parts:
$\frac{a + 1}{a} = k$
$\frac{a}{a} + \frac{1}{a} = k$
$1 + \frac{1}{a} = k$

From this, express $\frac{1}{a}$ in terms of $k$:
$\frac{1}{a} = k - 1$

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2: Simplify the target expression

Separate the target fraction in the same manner:
$\frac{a^2 - 1}{a^2} = \frac{a^2}{a^2} - \frac{1}{a^2} = 1 - \left(\frac{1}{a}\right)^2$

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3: Substitute the value

Replace $\frac{1}{a}$ with $(k - 1)$:
$1 - (k - 1)^2$

Expand the squared term using the algebraic identity $(x - y)^2 = x^2 - 2xy + y^2$:
$1 - (k^2 - 2k + 1)$
$1 - k^2 + 2k - 1$
$2k - k^2$

Factoring out $k$ gives the alternative form: $k(2 - k)$.