If S = $\sqrt{t^2 - 4t + 4}$, find t in terms of S

Analysis

The goal is to make $t$ the subject of the formula in the equation $S = \sqrt{t^2 - 4t + 4}$. To do this, we need to simplify the expression inside the square root and then isolate $t$.

Concept: Perfect Square Trinomials

In algebra, a perfect square trinomial follows the pattern: $a^2 - 2ab + b^2 = (a - b)^2$

Looking at our expression $t^2 - 4t + 4$:

• $t^2$ is the square of $t$.
• $4$ is the square of $2$.
• $-4t$ is $-2 \times (t) \times (2)$. This confirms that $t^2 - 4t + 4 = (t - 2)^2$.

Step-by-Step Solution

1. Substitute the simplified form into the equation: $S = \sqrt{(t - 2)^2}$

2. Simplify the square root: Since $\sqrt{x^2} = x$, the equation becomes: $S = t - 2$

3. Isolate $t$: To get $t$ by itself, add 2 to both sides of the equation: $S + 2 = t$

Therefore, $t = S + 2$.

Conclusion

Comparing our result ($t = S + 2$) with the given options, it matches option B.

However, in the context of JAMB multiple-choice questions, we assume the positive root, leading us to $t = S + 2$.*

Exam Tip

Whenever you see a quadratic expression inside a square root in an exam, always check if it is a perfect square first. It will save you time and prevent you from trying to use the complex quadratic formula unnecessarily!