# How do I solve this equation by completing the square?

ax^2 + bx + c= 0

Update:

yall are the sweetest, thanks boos xoxo

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• Your teacher is trying to see if you understand how completing the square works - by not using actual numbers. PLUS when we've done it, there's a bonus...

1) Move the "number" over to the right

ax^2 + bx = -c

2) Divide by the "number" multiplying the x^2 (the coefficient)

x^2 +(b/a) x = -c/a

3) Take half of the coefficient (don't forget the sign!) of the x-term, and square it. Add this square to both sides of the equation

The x is multiplied by b/a, so half of that is (b/2a). Squaring it, we get b^2/(4a^2)

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

4) Convert the left to a square

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

5) Take the square root of both sides

x + b/(2a) = ±√[b^2/(4a^2)-c/a]

6) Lets put the left hand side over a common denominator of 4a^2

x + b/(2a) = ±√[b^2-4ac/4a^2]

Now we can take that out of the square root

x + b/(2a) = ±√[b^2-4ac]/2a

x = -b/(2a)±√[b^2-4ac]/2a

x = [-b ±√[b^2-4ac]/(2a)

Recognize it ? Its the quadratic formula!

• set c off to the right side out of the way, and factor out a.

a(x² + b/a x + ....) = -c

now add half of the x coefficient, squared, inside the parentheses,

and an equal amount to the right side.

a[ x² + (b/a)x + (b / 2a)² ] = -c + a( b / 2a)²

a[ x² + b/a x + b² / 4a² ] = -c + ab² / 4a²

and rewrite left as a squared binomial while simplifying right

a[ x + b / 2a]² = -c + b² / 4a

give right a common denominator and add

a[ x + b / 2a]² = (-4ac + b²) / 4a

divide by a to isolate the square on left

(x + b / 2a)² = (b² – 4ac) / 4a²

square root each side

x + b / 2a = ±√(b² – 4ac) / 2a

and subtract to isolate x

x = -b / 2a ± √(b² – 4ac) / 2a

• Divide everything by the coefficient of the first term.

x² + (b/a)x + (c/a) = 0

Subtract the third term from both sides.

x² + (b/a)x = -(c/a)

Add half of the second term coefficient squared to both sides.

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

That is completing the square, and it leaves the left-hand side with this:

(x + b/2a)² = -c/a + b²/4a²

Multiply the first term on the right-hand side by 4a/4a for a common denominator, then simplify.

(x + b/2a)² = -4ac/4a² + b²/4a²

(x + b/2a)² = (b² - 4ac) / 4a²

Take the square root of both sides.

x + b/2a = ±√(b² - 4ac) / 2a

Now, subtract b/2a from both sides, giving you the Quadratic Formula

x = [-b ± √(b² - 4ac)] / 2a