Completing the square among other things

[scaez]

I need to know how to complete the square.. every website I go to about this is completely complicated.. my math teacher does it like so:
y = x^2 + 6x -7
y = x^2 + 6x + 3^2 - 9 - 7
y = (x + 3)^2 - 16

that I understand but..

if it were to say y = 2x^2 + 6x - 7

that's where I get confused because of the 2 infront of the x^2

I appreciate any help... -scaez

 

[cookiemonster]

Factor it out.

y = 2(x^2 + 3x + 7/2)

Leave the 2 out front and complete the square of x^2 + 3x + 7/2.

cookiemonster

 

[HallsofIvy]

(x+a)²= x²+ 2ax+ a².

To complete a square such as x²+ 3x, compare with that: we must have
2ax= 3x or a= 3/2. That gives a²= 9/4. We complete the square in x²+ 3x by adding 9/4: x²+ 3x+ 9/4= (x+ 3/2)².

I would be inclined to complete the square in 2x²+ 6x+ 7 by factoring the 2 out of the first two terms only: 2(x²+ 3x)+ 7. As before, to complete the square, add and subtract 9/4: 2(x²+ 3x+ 9/4- 9/4)+ 7/2. Since I ignored the "2", both 9/4s are inside the parentheses. I'll need to take -9/4 out of the parentheses and multiply by 2:
2x²+ 6x+ 7= 2(x²+ 3x+ 9/4- 9/4)+ 7= 2(x²+ 3x+ 9/4)- 9/2+ 7= (x+ 3/2)²+ 5/2.

 

[jcsd]

If yu like you can also put the factor back in the brackets:

y = 2x^2 + 3x + 7 = 2(x + 3/2)^2 + 5/4 = (sqrt(2)x + 3/sqrt(2))^2 + 5/4

 

[JonF]

You start with a polynomial

ax^2 + bx + c = 0

You factor “a” out of any terms with x

$$a(x^2 + \frac{b}{a}x) + c = 0$$

Then you divide b/a by ½ and square it.

So $$(\frac{b}{2a}})^2$$

Then you add that number inside the parenthesizes and subtract it multiplied by “a” (to keep the equation balanced)

$$a(x^2 + \frac{b}{a} x + (\frac{b}{2a})^2 ) + c -a{\frac{b}{2a}}^2 = 0$$

For example if you wanted to do the completing the square method for $$2x^2 + 16x +17 = -15$$

First you factor out a
$$2(x^2 + 8x) +17 = -15$$

then divide your b/a term by ½ and square it
(8/2)^2 = 16

add it to the inside and subtract it from the outside (multiply the outside one by “a”)

$$2(x^2 +8x + 16) - 32 + 17 = -15$$

factor and simplify

$$2(x+4)(x+4) = 0$$

x=-4

 

[geometer]

Here's another, maybe simpler way to look at it:

Starting with the general quadratic equation: (note - I lost my superscripts in translation here - I hope this doesn't confuse you!)

ax2 + bx+ c = 0

Subtract c from both sides:

ax2 + bx = -c

Divide both sides by a (we can do this since we know a does not equal 0. If it did, we wouldn’t have a quadratic equation)

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

Now comes the part where most people get confused. But, for now we’ll just do this. I’ll explain it in a second. Divide b/a by 2, and add the square of that to the right hand side of the equation. Then, change the left hand side of the equation so that it looks like this:

(x +b/2a)2 = b2/4a2 – c/a

Ok, all this comes from examining what happens when we square a binomial:
Consider:

(a + d) = 0

Squaring both sides:

(a+d)2 = 0

Which gives us: (Use the FOIL method – First, Outer, Inner, Last)

a2 + 2ad + d2 = 0

Notice the factor of 2 in the middle term on the left. So, if we want the middle term in our original equation (x2 + (b/a)x = -c/a) to come out right, the second term in the binomial we want to square has to be b/2a; half of b/a Now also notice that if we square (x+ b/2a) we will get a final term of b2/4a2 which didn’t appear in our original equation. But that’s easily fixed – just add it to the right hand side also.

So, moving on, we can add the two fractions on the right hand side of our new equation and get:

(x + b/2a)2 = (b2 – 4ac)/2a

And you have completed the square!

Note, that if you solve this for x you get the famous general solution for the quadratic equation.