Complete the steps of the derivation of the quadratic formula.

Question

Step 1: (x+b/2a)^2-b^2-4ac/4a^2=0

Answer 1

Step 2: Expand the left side of the equation:

(x^2 + 2bx + b^2)/(4a^2) - b^2 - 4ac/4a^2 = 0

Step 3: Multiply every term by 4a^2 to get rid of the fractions:

x^2 + 2bx + b^2 - 4a^2b^2 - 16ac = 0

Step 4: Rearrange terms to isolate x^2 term:

x^2 + 2bx = 4a^2b^2 + 16ac - b^2

Step 5: Complete the square by adding (b/2a)^2 to both sides:

x^2 + 2bx + (b/2a)^2 = 4a^2b^2 + 16ac - b^2 + (b/2a)^2

Step 6: Factor the left side:

(x + b/2a)^2 = 4a^2b^2 + 16ac - b^2 + b^2/4a^2

Step 7: Simplify the right side:

(x + b/2a)^2 = 4a^2b^2 + 16ac

Step 8: Take the square root of both sides:

x + b/2a = ±√(4a^2b^2 + 16ac)

Step 9: Simplify further:

x = (-b ± √(b^2 - 4ac))/(2a)

This is the quadratic formula.