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The area of a triangle is 12π₯ squared plus 38π₯ plus 28 centimeters squared, and its base is two π₯ plus four centimeters.
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Find an expression for its height.
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We now know that we can find the area of a triangle by multiplying the height times the base and then dividing by two, but we wanna rewrite this expression so that it says β equals something.
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How do we find the height if we know the area and the base?
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To find that, weβll need to isolate the β to get it by itself.
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Our first step would be to multiply by two on both sides.
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Two divided by two is one; it cancels out.
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On the right side, this leaves us with height times the base equals two times the area.
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From there we can divide both sides of our equation by the base.
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π divided by π equals one, so two times the area divided by the base equals the height.
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Letβs take this formula and plug in what we know.
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We know that the area equals 12π₯ squared plus 38π₯ plus 28 centimeters squared, and our base equals two π₯ plus four centimeters.
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We notice that we have centimeters squared in the numerator and centimeters in the denominator.
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One of the sets of centimeters cancel out, and that reminds us that our height will be given in centimeters not in centimeters squared.
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We wonβt change anything about our numerator.
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But for our denominator, I see a common factor of two.
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If I take out the factor of two, weβll reduce the expression to two times π₯ plus two.
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And then we notice that we have a two in our numerator and in our denominator.
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Two divided by two equals one; these cancel each other out.
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But how would we divide 12π₯ squared plus 38π₯ plus 28 by π₯ plus two?
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We need to use something called polynomial long division.
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From here we ask the question what can we multiply π₯ by two to equal 12π₯ squared.
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If I multiply 12π₯ by π₯, weβll get π₯ squared.
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But we have to multiply our 12π₯ by that second term as well, so weβll have to multiply 12π₯ by two.
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From here, weβll treat the problem like any other long division problem, and weβll subtract.
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12π₯ squared minus 12π₯ squared equals zero; 38π₯ minus 24π₯ equals 14π₯.
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We bring down our next term.
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Now we ask that same question again: what can we multiply π₯ by to equal 14π₯?
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Itβs 14; 14 times π₯.
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We canβt forget to multiply by that second term; 14 times two equals 28.
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We do some more subtraction here; 14π₯ minus 14π₯ equals zero; 28 minus 28 equals zero.
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Now we have nothing remaining.
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12π₯ plus 14 is what we get when we divide two times the area by the base, and we can say that the height of this triangle equals 12π₯ plus 14 centimeters.