Can somebody give a step by step guide on solving quadratics and completing the square?

Question

Thanks

Answer 1

Let's start with an equation of the form: x^2 + bx + c = 0.
This means that x^2 + bx = -c.

Consider the formula ( x + k )^2 = x^2 + 2k x + k^2. Every perfect square is in this form.
We want to change x^2 + bx into a perfect square. That means we need to determine what to add to put x^2 + bx into this form.

By equating coefficients: bx = 2k x. That means k = (b/2)
Thus k^2 = (b/2)^2.

Adding (b/2)^2 to both sides of our earlier equation we arrive at:

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

We added this to convert the left hand side into the form of a perfect square. Thus we can factor it as a square.

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

Take the square root of both sides ( for now just assume that the right hand side is positive ).

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

now solve for x:

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

Example:

Solve x^2 - 6x - 4 = 0

Put into the proper form: x^2 - 6x = 4
Determine k: b = -6 so k = -3
Determine k^2: k = -3 so k^2 = 9
Add k^2 to both sides: x^2 - 6x + 9 = 13
Factor left hand side: (x - 3)^2 = 13 this is ALWAYS (x+k)^2
Square root: x-3 = +/- sqrt( 13 )
Solve for x: x = 3 +/- sqrt( 13 )


For a more general equation: a x^2 + b x + c = 0
First divide through by a to get: x^2 + (b/a) x + (c/a) = 0
This is now in the earlier form so use that method will solve it.

Good Luck!

Answer 2

ax^2+bx+c=0

First divide everything by a:

x^2+(b/a)x+c/a=0

Now add (b/2a)^2 to both sides (this will make sense in a bit):

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

Now notice that x^2+(b/a)x+(b/2a)^2=(x+b/2a)^2

So we now have:

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

Take the square root of both sides:

x+b/2a=±√[(b^2-4ac)/(4a^2)]= ±√(b^2-4ac)/(2a)

subtracting b/2a from both sides gives:

x=(-b±√(b^2-4ac))/(2a). This is known as the quadratic equation. You can now put any value of a, b, c into it (a≠0) and you will have the solutions to your quadratic polynomial.

An example of completing the square for 4x^2+16x-20=0

First divide by 4:

x^2+4x-20=0

notice that 4/2=2 and 2^2=4, so take that from the 20:

x^2+4x+4-4-20=x^2+4x+4-24=0.

Notice that x^2+4x+4=(x+2)^2:

(x+2)^2-24=0 or (x+2)^2=24

Square root of both sides:

x+2=±√(24)=±2√6

Thus x=-2±2√6 are the solutions.

Answer 3

qhen doing a quadratic equation, use the F O I L method of multiplying. that stands for FIRSTS OUTERS INNERS LASTS.