Completing the square and fractional quadratics

Complete a square without changing the expression

x² + 6x + 2 = (x + 3)²-9 + 2 = (x + 3)²-7. The extra 9 inside the square is balanced by subtracting 9 outside.

For the equation x² + 6x + 2 = 0, obtain (x + 3)² = 7, then x = -3± square root of 7. Both square-root signs are needed. For the function y = x² + 6x + 2, the same form shows a minimum value −7 at x = -3, because a real square is non-negative. Rewriting the function alone does not set its output to zero.

Fractional equations may become quadratics

For 6/(x + 4) = x + 3, first exclude x = -4. Multiplying gives 6 = (x + 4)(x + 3), so x² + 7x + 6 = 0. Factor to obtain (x + 1)(x + 6) = 0, giving x = -1 or x = -6. Both are allowed and both satisfy the original.

An equation such as 1/(x-2) + 2/(x-3) = 5 first excludes 2 and 3. Clearing denominators gives 3x-7 = 5(x-2)(x-3), or 5x²-28x + 37 = 0. The formula then gives (14± square root of 11)/5, neither excluded. Keep the original domain with the answer.

Check your understanding 1 Preserve the constant

Rewrite x²-8x + 5 as a completed square and solve x²-8x + 5 = 0.

Your answer or drawing on paper

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x²-8x + 5 = (x-4)²-11. Therefore x = 4± square root of 11. Expanding (x-4)² shows the added 16, which must be removed before adding 5.

A graph gives a different form of evidence

Sketching y = x²-8x + 5 locates a minimum at (4, −11) and two crossings near 0.68 and 7.32. Use those estimates to check the exact roots, not to replace exact answers when they are requested.