Question

A portion of the Quadratic Formula proof is shown. Fill in the missing reason.

Answer 1

The quadratic equation ax²+bx+c = 0 is given and this is illustrated below.

How to illustrate the equation?

ax²+bx+c = 0

Step 1: Subtract c from both sides

ax²+bx+c-c = 0-c

ax²+bx = -c

Step 2: Divide both sides of the equation by a

ax²/a + bx/a = -c/a

x² + bx/a = -c/a

Step 3: Complete the square and add the quantity (b/2a)² times a squared to both sides

x² + bx/a +  (b/2a)² = -c/a +  (b/2a)²

Step 4: Square the quantity b/2a on the right side of the equation

x² + bx/a +  (b/2a)² =  -c/a +  b²/4a²

Step 5: Find a common denominator on the right side of the equation which is 4a²

x² + bx/a +  (b/2a)² =  -4ac/4a² +  b²/4a²

Step 6: Add the fractions together on the right side of the equation

x² + bx/a +  (b/2a)² =  (-4ac+  b²)/4a²

Step 7: The equation on the left is to be written as a perfect square as shown

(x+b/2a)² =  (-4ac+  b²)/4a²

Step 8: Take the square root of both sides

√(x+b/2a)² = √ (-4ac+  b²)/4a²

(x+b/2a) =  √(-4ac+  b²)/2a

Step 9: subtract b/2a from both sides

x+b/2a - b/2a =  -b/2a + √(-4ac+  b²)/2a

x =  -b/2a + √(-4ac+  b²)/2a

Step 10: Add the fractions together on the right-hand side

x =  -b±√(-4ac+  b²)/2a

This will then gives the required equation.

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