Problem 122 - College Algebra

Question

How is the quadratic formula derived?

Step-by-Step Solution

Answer

The quadratic formula \( x = [-b ± sqrt(b^2 - 4ac)] / 2a \) is derived by completing the square on a general quadratic equation and then applying square root property and simplifying.

Step 1: Write the Quadratic Equation

Write down the general quadratic equation \( ax^2 + bx + c = 0 \)

Step 2: Make 'a' equal to 1

Divide all terms by ‘a’ to make coefficient of square term equal to 1. \( x^2 + (b/a)x + c/a = 0 \)

Step 3: Move the Constant on the Other Side

Move the constant \( c/a \) to another side of the equation to make it \( x^2 + (b/a)x = -c/a \)

Step 4: Complete the Square

Complete the square on the left side of the equation and balance it by adding the same value to the right side. \( x^2 + (b/a)x + (b/2a)^2 = -c/a + (b/2a)^2 \)

Step 5: Application of Square Root Property

Apply square root property to solve for \( x \). \( x = [-b ± sqrt{ (b^2 - 4ac) / (4a^2) }] / 2a \)

Step 6: Simplify

Simplify the above expression to obtain the quadratic formula. \( x = [-b ± sqrt{b^2 - 4ac} ] / 2a \)