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Worked example: completing the square (leading coefficient ≠ 1)

Sal solves the equation 4x^2+40x-300=0 by completing the square. Created by Sal Khan and Monterey Institute for Technology and Education.

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• The process doesn't seem that confusing but if the result is not a perfect square than what's next? an example is x squared +16x+57=0 the result I go was x+8=7 but this doesn't check... I need help please
• 1.Subtract 57 from both sides, which will give you x squared+16x=-57.2.Complete the square: x squared+16x+64=7.3.Factor the left side of the equation: (x+8)squared=7.4.Square root both sides of the equation: x+8= positive or negative square root of 7.5. Split the equations into: x+8=positive square root of 7 and x+8=negative square root of 76. Solve both equation!
• Where did the term "coefficient" come from?
• 1655-65; from New Latin coefficiēns, from Latin co- together + efficere to effect.
• So the practice after this video only managed to completely confuse me. Sometimes you divided everything by the leading coefficient, sometimes you don't divide the last term by the leading coefficient, sometimes you multiple the squared middle term by the leading coefficient. The explanations suck as to why you do this and not that, so can someone help me out please?

Example:
2x^2 + 3x - 2 = 0
2 (x^2 +3/2x - 1)

Then divide the middle term to get 3/4, then I subtract that term squared from -1 to get -1 - 9/16, to which I got 25/16 = (x+3/4)^2 or 2(x+3/4)^2 - 25/16

But the hint for the equation showed this process instead:
2(x^2+3/2x) - 2
(x + 3/4)^2 - 2 - 2* 9/16
2(x+3/4)^2 - 25/8

Why didn't they divide the 2 term by 2 in the beginning? And why did they times the added term by 2 at the end? Looking back at it, I'm thinking they multiplied the last term by 2 to make it even with the equation in the paratheses, but I've also seen equations when the term isn't multiplied by the leading coeffiecient. Help?
• 5 years late, but I believe I know the answer:
The example question's answer was supposed to be written in vertex form, and the method they used for doing this was by factoring only the first two terms like so:
2x² + 3x -2 = y
2(x² + 3/2x) - 2 = y
Then to make a perfect square, they added 9/16 in the parenthesis to make 2(x + 3/4)². But to retain the exact value of the equation, 9/16 multiplied by 2 (factored from the parenthesis) must be subtracted.
2(x + 3/4)² - 2 - 2 * 9/16 = y
2(x + 3/4)² - 25/8 = y
The process you tried by making y = 0 also could have worked, but there was an error in the last step.
25/16 = (x + 3/4)²
2 * 25/16 = 2(x + 3/4)²
2(x + 3/4)² - 25/8 = 0
Hope this helps :)
• how can my calculator solve this problem using the quadratic formula? since -b+square root (b²-4ac) would be -40+ square root of (40²-4.4.(-300)) which is equal to -40 + square root of (1600 - 4800). wouldn't that be taking the square root of a negative number? or am I using the wrong order of operations?
• Hi Dimitri,
It looks like your are forgetting to multiply the two negatives together. If you have:
40^2 - (4)(4)(-300)
that will give you
1600 - (-4800)
which equals
1600 + 4800 = 6400
Hope that helps!
• The practice questions are not explained well by this video.
How does f(x)=2x^2+3x−2 end up as f(x)=2(x+3/4​)^2−25/8​ ?

Is there not a simpler answer?
• There is no simpler answer. First, factor out a 2 in first 2 terms to get f(x)=2(x^2+3/2 x) - 2. To complete the square, divide 3/2 by 2 to get 3/4, then square to get 9/16. This would give f(x)=2(x^2+3/2 x +9/16-9/16)-2. Taking the -9/16 out, you have to multiply this by 2 to get -9/8, thus ending up with f(x)=2(x^2+3/2x +9/16)-2-9/8, finding common denominator of 8 and combining gives -16/8-9/8=-25/8. This gives your final result.
• I didn't factor out the 4 at the beginning and ended up with a funky answer with a square root of 7. I understand how to do it properly and got the same answer of x=5 or -15 as Sal did when I factored out the 4 first, but just don't fully get why. Help? Thanks!
• 4x^2 + 40x - 300 = 0 so we have 4x^2 + 40x = 300
Since (ax +b)^2 = a^2x^2 + 2abx + b^2, that means a = 2, so the middle term is 2 • 2 b = 40, so b = 10 and b^2 = 100
completing the square (2x + 10)^2 = 300 +100
(2x + 10)^2 = 400, take the square root of both sides, adding +/- on right
2x +10 = +/- 20, 2x = 20 -10 or 2x = -20 - 10
2x = 10 or 2x = -30
x = 5 or x = -15
Without factoring, there are a whole lot of places to mess up, probably one of the most common mistakes is getting b incorrect. Find where you messed up.
• The practice problems all have fractions, and the video doesn't cover them. For example, how do you solve g(x)=2x^2−7x+5
• There are some examples in the questions and answers already posted for this lesson. You can also find some great videos on the internet. The process doesn't change. You just need to work with the fractions.

g(x)=2x^2−7x+5
A) Factor 2 from first 2 terms: g(x)=2(x^2−7/2 x)+5
B) Complete the square: Divide -7/2 by 2 to get -7/4. Square it (-7/4)^2 = 49/16. Note: the -7/4 will be used in the factors later on.
C) Add & subtract 49/16: g(x)=2(x^2−7/2 x +49/16 - 49/16)+5
D) Multiply -49/16 by the 2 in front of the parentheses so it can come out of the parentheses to get: g(x)=2(x^2−7/2 x +49/16) -49/8+5
E) Simplify the constant term by adding using an LCD to get: -49/8+5 = -49/8+40/8 = -9/8
F) Factor the parentheses to show the square:
g(x) = 2(x-7/4)^2 -9/8

Hope this helps.
• Why the 1st coefficient has to be 1?
• The pattern used to Complete the Square only works if the coefficient of X^2 is = 1. If the coefficient is not 1, dividing the middle term by 2 and squaring will not create the correct values.