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13 Jan 2026

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Comprehensive Guide to Completing the Square

🎸🦕🕸️𝔱𝔥𝔢𝔬🕸️🦕🎸

@gh0styb0i

Completing the square is a powerful algebraic technique that transforms... Show more

MATHEMATICS-Completing the
Equations
NOTES/WORK
Square
x²+ bx = (x + 2)² - (1/2) ²
Proof of the quadratic formula
ax² + bx+c=0
x² + bx += =

Completing the Square and the Quadratic Formula

Ever wondered where the quadratic formula actually comes from? It's not just magic - it's derived using completing the square, and once you understand this connection, quadratic equations become much less intimidating.

The basic pattern for completing the square is: x² + bx = $x + b/2$ ² - $b/2$ ². This formula lets you turn any quadratic expression into a perfect square plus or minus a constant. The key is taking half of the coefficient of x, squaring it, and both adding and subtracting it.

To prove the quadratic formula, we start with ax² + bx + c = 0 and divide everything by 'a' to get x² + $b/a$ x + c/a = 0. Then we complete the square by adding $b/2a$ ² to both sides, which gives us $x + b/2a$ ² = $b² - 4ac$ /4a². Taking the square root and rearranging leads directly to the familiar formula: x = $-b ± √(b² - 4ac)$ /2a.

The practice questions show different types: simple cases like x² + 6x and x² - 10x, plus trickier ones where you need to factor out coefficients first, like 2x² + 9x and 3x² - 5x + 7.

Quick Tip: Always remember to halve the coefficient of x, then square that result - this is the number you add and subtract when completing the square.

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1 page

Comprehensive Guide to Completing the Square

🎸🦕🕸️𝔱𝔥𝔢𝔬🕸️🦕🎸

@gh0styb0i

Completing the square is a powerful algebraic technique that transforms quadratic expressions into a more useful form. It's the foundation behind the quadratic formula and helps you solve equations that might otherwise seem impossible to crack.

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Completing the Square and the Quadratic Formula

The practice questions show different types: simple cases like x² + 6x and x² - 10x, plus trickier ones where you need to factor out coefficients first, like 2x² + 9x and 3x² - 5x + 7.

Quick Tip: Always remember to halve the coefficient of x, then square that result - this is the number you add and subtract when completing the square.