Example 3 Use
the quadratic formula to solve each of the following equations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
The important part here is to make sure that before we
start using the quadratic formula that we have the equation in standard form
first.
(a)
So, the first thing that we need to do here is to put the
equation in standard form.
At this point we can identify the values for use in the
quadratic formula. For this equation
we have.
Notice the “” with c. It is important to make sure that we carry
any minus signs along with the constants.
At this point there really isn’t anything more to do other
than plug into the formula.
There are the two solutions for this equation. There is also some simplification that we
can do. We need to be careful
however. One of the larger mistakes at
this point is to “cancel” two 2’s in the numerator and denominator. Remember that in order to cancel anything
from the numerator or denominator then it must be multiplied by the whole
numerator or denominator. Since the 2
in the numerator isn’t multiplied by the whole denominator it can’t be
canceled.
In order to do any simplification here we will first need
to reduce the square root. At that
point we can do some canceling.
That’s a much nicer answer to deal with and so we will
almost always do this kind of simplification when it can be done.
[Return to Problems]
(b)
Now, in this case don’t get excited about the fact that
the variable isn’t an x. Everything works the same regardless of the
letter used for the variable. So,
let’s first get the equation into standard form.
Now, this isn’t quite in the typical standard form. However, we need to make a point here so
that we don’t make a very common mistake that many student make when first
learning the quadratic formula.
Many students will just get everything on one side as
we’ve done here and then get the values of a, b, and c based upon position. In other words, often students will just
let a be the first number listed, b be the second number listed and then
c be the final number listed. This is not correct however. For the quadratic formula a is the coefficient of the squared
term, b is the coefficient of the
term with just the variable in it (not squared) and c is the constant term.
So, to avoid making this mistake we should always put the quadratic
equation into the official standard form.
Now we can identify the value of a, b, and c.
Again, be careful with minus signs. They need to get carried along with the
values.
Finally, plug into the quadratic formula to get the
solution.
As with all the other methods we’ve looked at for solving
quadratic equations, don’t forget to convert square roots of negative numbers
into complex numbers. Also, when b is negative be very careful with the
substitution. This is particularly
true for the squared portion under the radical. Remember that when you square a negative
number it will become positive. One of
the more common mistakes here is to get in a hurry and forget to drop the
minus sign after you square b, so
be careful.
[Return to Problems]
(c)
We won’t put in quite the detail with this one that we’ve
done for the first two. Here is the
standard form of this equation.
Here are the values for the quadratic formula as well as
the quadratic formula itself.
Now, recall that when we get solutions like this we need
to go the extra step and actually determine the integer and/or fractional
solutions. In this case they are,
Now, as with completing the square, the fact that we got
integer and/or fractional solutions means that we could have factored this
quadratic equation as well.
[Return to Problems]
(d)
So, an equation with fractions in it. The first step then is to identify the LCD.
So, it looks like we’ll need to make sure that neither or is in our answers so that we don’t get
division by zero.
Multiply both sides by the LCD and then put the result in
standard form.
Okay, it looks like we’ve got the following values for the
quadratic formula.
Plugging into the quadratic formula gives,
Note that both of these are going to be solutions since neither
of them are the values that we need to avoid.
[Return to Problems]
(e)
We saw an equation similar to this in the previous section
when we were looking at factoring equations and it would definitely be easier
to solve this by factoring. However,
we are going to use the quadratic formula anyway to make a couple of points.
First, let’s rearrange the order a little bit just to make
it look more like the standard form.
Here are the constants for use in the quadratic formula.
There are two things to note about these values. First, we’ve got a negative a for the first time. Not a big deal, but it is the first time
we’ve seen one. Secondly, and more
importantly, one of the values is zero.
This is fine. It will happen on
occasion and in fact, having one of the values zero will make the work much
simpler.
Here is the quadratic formula for this equation.
Reducing these to integers/fractions gives,
So we get the two solutions, and . These are exactly the solutions we would
have gotten by factoring the equation.
[Return to Problems]
