We are asked to solve
the quadratic equation
by completing the square.
The first step is to write the equation
in the form shown here
which for our equation is
in the form x-squared
minus 5x plus some constant
that will make the
trinomial a perfect square,
equals negative four, and
then plus the same constant
we added to the left.
Next if a, the coefficient
of x-squared is not one,
we divide both sides by a.
Notice here a is equal to one,
and therefore we can skip to step three.
Step three, we add the
square of b divided by two
with a square of one-half b
to both sides of the equation.
This will make the left side
a perfect square trinomial.
Well b is the coefficient of x
and therefore b is equal to negative five.
So we need to add the square
of negative five divided
by two to both sides of the equation.
The square of negative five-halfs
is equal to 25-fourths.
We now add 25-fourths to
both sides of the equation.
We add it to both sides
to maintain the equality.
Step three is to factor the left side
which is now a perfect square trinomial.
It's not obvious here, but it is.
Show some work if this does factor,
it will factor into two binomial factors.
'Cause the first term is x-squared,
we have an x in the first
position of both binomials.
The second terms of the binomials,
the second terms of the
binomial factors will be
the factors of 25-fourths
that add to negative five
which is not an easy question,
but the factors will always be
the two factors we multiplied
to get the 25-fourths.
We actually multiplied
negative five-halves by itself
to get the 25-fourths.
So we already know that
negative five-halves times
a negative five-halves
is equal to 25-fourths.
Let's just verify.
Negative five-halves plus
negative five-halves is
equal to negative five.
This sum is equal to a negative 10 halves
which is equal to negative five.
So we have two binomial
factors of x minus five-halves.
Every time I complete the square,
b divided by two or half of
b, this term here, will be
the term in both binomial factors.
Now on the right side,
let's determine the sum.
We have negative four or
negative four over one
plus 25-fourths.
The least common denominator is four.
We multiple the numerator and denominator
of negative four over one by four.
This gives us negative 16
fourths plus 25 fourths
which is equal to nine-fourths.
So the right side
simplifies to nine-fourths.
And notice how we do have
a perfect square trinomial
because we have two
equal binomial factors.
Let's write this as the
quanity x minus five-halves
squared equals nine-fourths.
The next step is to square root both sides
of the equation and solve for x.
And when square rooting both
sides of the equation is
important to remember, we include a plus
or minus on the right so
that we get both solutions.
On the left, the square
root undoes the squaring
and we're left with the
quantity x minus five-halves.
And on the right, three
squared is equal to nine.
Two squared is equal to four.
And therefore the squareroot
of nine-fourths is equal
to three-halves.
The right side is now just
plus or minus three-halves.
And then to solve for x we
add five-halves to both sides.
Negative five-halves
plus five-halves is zero.
We now have x equals five-halves plus
or minus three-halves.
Now we need to find the sum or difference
to give the two solutions.
One solution is x equals
five-halves plus three-halves.
And the other solution is
x equals five-halves minus
three halves.
Going back up to the sum, five-halves plus
three-halves is eight-halves
which simplifies to four.
And five-halves minus
three-halves is two-halves
which simplifies to one.
So our two solutions are x equals one
or x equals four.
This is how we solve the given equation
by completing the square.
But I do want to point
out, it would have been
much easier to solve this equation
by setting it equal to zero
and then just factoring
if we were given this option.
Let's quickly show this.
To set the equation equal to zero,
we would add four to both sides.
Simplifying we now have the equation
x-squared minus 5x plus four equals zero.
Now we can factor the
trinomial on the left.
If it does factor, it factors
into two binomial factors.
First term is x-squared
which is equal to x times x
and because leading coefficient is one,
we now find the factors of positive four
that add to negative five
which are negative four
and negative one.
Which means one binomial
factor is x minus one
and the other is x minus four.
And notice how the factor
of x minus one is equal
to zero when x equals one.
And the factor of x minus
four is equal to zero
when x equals four.
So of course we get the same solution
much faster, but we were
asked to complete the square.
I hope you found this helpful.
