WEBVTT
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Consider the identity two π₯ minus five multiplied by π₯ plus three plus ππ₯ plus π is identically equal to two π₯ squared plus four π₯ minus three.
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Work out the values of π and π.
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An identity is a statement which is always true no matter what the value of the variable, in this case π₯, takes.
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Itβs denoted by this sign here, an equal sign with an extra horizontal line to indicate that the statement is always true.
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Weβre asked to work out the values of π and π.
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And to do this, weβll begin by expanding and simplifying the left-hand side of this identity.
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Weβll use the FOIL method to expand the brackets.
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First, we multiply two π₯ by π₯, giving two π₯ squared.
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Then we multiply the outer terms together.
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Two π₯ multiplied by three gives positive six π₯.
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Next, the inner terms, negative five multiplied by π₯ gives negative five π₯.
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And finally, the last terms, negative five multiplied by three gives negative 15.
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We can then bring down the plus ππ₯ plus π from the previous line and also the right-hand side of this identity.
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Next, we can simplify the like terms in our expansion.
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Positive six π₯ minus five π₯ just leaves positive one π₯ or π₯.
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So we have two π₯ squared plus π₯ minus 15 plus ππ₯ plus π is identically equal to two π₯ squared plus four π₯ minus three.
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Now weβre actually going to go a little bit further with our simplification because we still have like terms.
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We have plus π₯ and then plus ππ₯.
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We can combine these two terms together with a coefficient of one plus π, because when we expand this bracket, weβll get π₯ plus ππ₯.
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Weβll also combine our constant term.
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We have negative 15 plus π, which we can write as π minus 15.
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We then bring down the right-hand side of the identity.
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Now hereβs a key fact that we need to know in order to answer this question.
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If this statement is an identity, then it means that the coefficient of π₯ squared, π₯, and the constant term must be the same on both sides of this identity in order for it to be true for all values of π₯.
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We can see, for example, that the coefficient of π₯ squared is two on each side.
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In order to find the values of π and π, we need to compare the coefficients of π₯ and the constant term on the two sides of the equation.
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If we compare the coefficients of π₯ first of all, we have one plus π on the left side of this identity and four on the right.
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So we have the equation one plus π equals four, which we can solve to find the value of π.
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We just need to subtract one from each side, giving π equals three.
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Finally, we compare the constant terms.
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We have π minus 15 on the left and negative three on the right.
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So we have an equation that we can solve for π.
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We add 15 to each side of the equation, giving π equals 12.
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So by expanding, simplifying, and then comparing coefficients, we found the values of π and π.
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π is equal to three, and π is equal to 12.