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: xยฒ + bx = x+b/2ยฒ - b/2ยฒ. 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 axยฒ + bx + c = 0 and divide everything by 'a' to get xยฒ + b/ax + c/a = 0. Then we complete the square by adding b/2aยฒ to both sides, which gives us x+b/2aยฒ = b2โ4ac/4aยฒ. Taking the square root and rearranging leads directly to the familiar formula: x = โbยฑโ(b2โ4ac)/2a.
The practice questions show different types: simple cases like xยฒ + 6x and xยฒ - 10x, plus trickier ones where you need to factor out coefficients first, like 2xยฒ + 9x and 3xยฒ - 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.