What is the minimum value of the expression $(x - 3)^{2} + (x + 4)^{2}$ ?

medium MCQ 2 marks Class 8 · Mathematics · Squares and Square Roots

Answer

24.5

The correct answer is 24.5 — Step 5: Hence the minimum value is $24.5$ .

Step-by-step solution

Step 1: Expand the expression: $(x - 3)^{2} + (x + 4)^{2} = x^{2} - 6 x + 9 + x^{2} + 8 x + 16 = 2 x^{2} + 2 x + 25$ .
Step 2: Differentiate with respect to $x$ : $\frac{d}{d x} (2 x^{2} + 2 x + 25) = 4 x + 2$ .
Step 3: Set derivative to zero for minimum: $4 x + 2 = 0 \to x = - \frac{1}{2}$ .
Step 4: Substitute $x = - \frac{1}{2}$ into $2 x^{2} + 2 x + 25$ : $2 (\frac{1}{4}) + 2 (- \frac{1}{2}) + 25 = 0.5 - 1 + 25 = 24.5$ .
Step 5: Hence the minimum value is $24.5$ .

Common mistake: Not expanding correctly, missing cross terms or using wrong derivative

Concept tested: optimization using squares