4.3 Completing the Square: The Geometric Method

Step 1

Divide by 5: $x^2 - \frac{6}{5}x - \frac{2}{5} = 0$ .

Step 2

Move constant: $x^2 - \frac{6}{5}x = \frac{2}{5}$ .

Step 3

Find term to add: Coeff of x is $-6/5$ . Half is $-3/5$ . Square is $9/25$ .

Step 4

Add $9/25$ to both sides: $x^2 - \frac{6}{5}x + \frac{9}{25} = \frac{2}{5} + \frac{9}{25}$ .

Step 5

Factor LHS and simplify RHS: $(x - \frac{3}{5})^2 = \frac{10}{25} + \frac{9}{25} = \frac{19}{25}$ .

Step 6

Take square root: $x - \frac{3}{5} = \pm \frac{\sqrt{19}}{5}$ .

Step 7

Solve: $x = \frac{3}{5} \pm \frac{\sqrt{19}}{5} = \frac{3 \pm \sqrt{19}}{5}$ .

Final Answer

$x = \frac{3 \pm \sqrt{19}}{5}$

Input

Matrix

2×2

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