How do I solve this equation by completing the square?
I'm really confused because there are just variables..please help!
ax^2 + bx + c= 0
yall are the sweetest, thanks boos xoxo
- ireadlotsLv 68 years ago
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...
Just follow the normal steps:
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!
- PhiloLv 78 years ago
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
- Damon LyonLv 78 years ago
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