Video: AQA GCSE Mathematics Higher Tier Pack 4 β’ Paper 1 β’ Question 17

Consider the identity (2π₯ β 5) (π₯ + 3) + ππ₯ + π = 2π₯Β² + 4π₯ β 3. Work out the values of π and π.

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Video Transcript

Consider the identity two π₯ minus five multiplied by π₯ plus three plus ππ₯ plus π is identically equal to two π₯ squared plus four π₯ minus three. Work out the values of π and π.

An identity is a statement which is always true no matter what the value of the variable, in this case π₯, takes. Itβs denoted by this sign here, an equal sign with an extra horizontal line to indicate that the statement is always true.

Weβre asked to work out the values of π and π. And to do this, weβll begin by expanding and simplifying the left-hand side of this identity. Weβll use the FOIL method to expand the brackets. First, we multiply two π₯ by π₯, giving two π₯ squared. Then we multiply the outer terms together. Two π₯ multiplied by three gives positive six π₯. Next, the inner terms, negative five multiplied by π₯ gives negative five π₯. And finally, the last terms, negative five multiplied by three gives negative 15. We can then bring down the plus ππ₯ plus π from the previous line and also the right-hand side of this identity.

Next, we can simplify the like terms in our expansion. Positive six π₯ minus five π₯ just leaves positive one π₯ or π₯. So we have two π₯ squared plus π₯ minus 15 plus ππ₯ plus π is identically equal to two π₯ squared plus four π₯ minus three.

Now weβre actually going to go a little bit further with our simplification because we still have like terms. We have plus π₯ and then plus ππ₯. We can combine these two terms together with a coefficient of one plus π, because when we expand this bracket, weβll get π₯ plus ππ₯. Weβll also combine our constant term. We have negative 15 plus π, which we can write as π minus 15. We then bring down the right-hand side of the identity.

Now hereβs a key fact that we need to know in order to answer this question. 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₯.

We can see, for example, that the coefficient of π₯ squared is two on each side. 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. 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. So we have the equation one plus π equals four, which we can solve to find the value of π. We just need to subtract one from each side, giving π equals three.

Finally, we compare the constant terms. We have π minus 15 on the left and negative three on the right. So we have an equation that we can solve for π. We add 15 to each side of the equation, giving π equals 12. So by expanding, simplifying, and then comparing coefficients, we found the values of π and π. π is equal to three, and π is equal to 12.