Question Video: Solving Quadratic Equations by Factoring Perfect Squares Mathematics

Solve the equation 9π‘₯Β² + 30π‘₯ + 25 = 0 by factoring.

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Video Transcript

Solve the equation nine π‘₯ squared plus 30π‘₯ plus 25 equals zero by factoring.

This is an equation that contains a nonmonic quadratic, a quadratic with a coefficient of π‘₯ 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 π‘Ž 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 π‘₯ 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 30π‘₯ into 15π‘₯ and 15π‘₯. And so our quadratic expression is nine π‘₯ squared plus 15π‘₯ plus 15π‘₯ plus 25. We now individually factor the first two terms and the last two terms. The greatest common factor of nine π‘₯ squared and 15π‘₯ is three π‘₯. So factoring these first two terms, we get three π‘₯ times three π‘₯ plus five. Then the greatest common factor of our last two terms is five. And so when we factor 15π‘₯ plus 25, we get five times three π‘₯ plus five.

Notice now that we have a common factor of three π‘₯ plus five. So we’re going to factor that. Three π‘₯ plus five is multiplied by three π‘₯ and five. So that’s the other binomial. And our expression becomes three π‘₯ plus five times three π‘₯ plus five.

Now, of course, we’re solving the equation nine π‘₯ squared plus 30π‘₯ 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 π‘₯ plus five is equal to zero or three π‘₯ 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 π‘₯ plus five equals zero. We subtract five from both sides. So three π‘₯ is negative five. And then we divide through by three. So π‘₯ is equal to negative five-thirds. And so we see that the equation nine π‘₯ squared plus 30π‘₯ plus 25 equals zero has the solution π‘₯ 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 π‘₯ equals negative five-thirds into our original expression. And if we’d done that correctly, we’d find it’s equal to zero.

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