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