Solve the equation nine 𝑥 squared
plus 30𝑥 plus 25 equals zero by factoring.
This is an equation that contains a
nonmonic quadratic, a quadratic with a coefficient of 𝑥 squared is not equal to
one. This means the quadratic expression
is a little bit more difficult to factor than usual. We might notice that it’s a perfect
square, with 𝑎 and 𝑐 being square numbers. But if we didn’t spot this, we
could use trial and error or use the following method to factor.
In this method, the first thing
that we do is we multiply the coefficient of 𝑥 squared and the constant. Nine times 25 is 225. And so we look for two numbers
whose product is 225 and whose sum is 30. Well, 225 is a square number such
that 15 times 15 is 225. And we also know that the sum of 15
and 15 is 30.
Our next step then is to break the
30𝑥 into 15𝑥 and 15𝑥. And so our quadratic expression is
nine 𝑥 squared plus 15𝑥 plus 15𝑥 plus 25. We now individually factor the
first two terms and the last two terms. The greatest common factor of nine
𝑥 squared and 15𝑥 is three 𝑥. So factoring these first two terms,
we get three 𝑥 times three 𝑥 plus five. Then the greatest common factor of
our last two terms is five. And so when we factor 15𝑥 plus 25,
we get five times three 𝑥 plus five.
Notice now that we have a common
factor of three 𝑥 plus five. So we’re going to factor that. Three 𝑥 plus five is multiplied by
three 𝑥 and five. So that’s the other binomial. And our expression becomes three 𝑥
plus five times three 𝑥 plus five.
Now, of course, we’re solving the
equation nine 𝑥 squared plus 30𝑥 plus 25 equals zero. So let’s set this equal to
zero. And we know that for the product of
these two numbers to be equal to zero, either one or other number must itself be
equal to zero. So we see that three 𝑥 plus five
is equal to zero or three 𝑥 plus five is equal to zero. In fact, these are the same
equation and they’ll yield the same result.
So we’re just going to solve the
equation three 𝑥 plus five equals zero. We subtract five from both
sides. So three 𝑥 is negative five. And then we divide through by
three. So 𝑥 is equal to negative
five-thirds. And so we see that the equation
nine 𝑥 squared plus 30𝑥 plus 25 equals zero has the solution 𝑥 equals negative
five-thirds. We can say that our equation has
two equal roots or a repeated root.
And remember, we could actually
check our working by substituting 𝑥 equals negative five-thirds into our original
expression. And if we’d done that correctly,
we’d find it’s equal to zero.