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Assume that a and b are integers such that a^2 - 4b - 2 = 0 9 (a) Prove that a is even - AQA - A-Level Maths Pure - Question 9 - 2022 - Paper 3

Question 9

Assume that a and b are integers such that a^2 - 4b - 2 = 0 9 (a) Prove that a is even. 9 (b) Hence, prove that 2b + 1 is even and explain why this is a contradic... show full transcript

Worked Solution & Example Answer:Assume that a and b are integers such that a^2 - 4b - 2 = 0 9 (a) Prove that a is even - AQA - A-Level Maths Pure - Question 9 - 2022 - Paper 3

Step 1

Prove that a is even.

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Answer

To prove that $a$ is even, we start with the equation given:

$a^2 - 4b - 2 = 0$

Rearranging it, we find:

$a^2 = 4b + 2$

Here, the term on the right side ( $4b + 2$ ) is clearly even because it is the sum of an even number ( $4b$ , which is a multiple of 4) and another even number (2).

Since the left side ( $a^2$ ) is also equal to an even number, $a$ itself must be even. This is because the square of an odd number is odd, thus contradicting the equality if $a$ were odd. Therefore, we conclude that:

$a$ is even.

Step 2

Hence, prove that 2b + 1 is even and explain why this is a contradiction.

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Answer

Given that $a$ is even, we can rewrite $a$ as $a = 2k$ for some integer $k$ . Substituting this back into our original equation, we have:

$(2k)^2 - 4b - 2 = 0$

Expanding this yields:

$4k^2 - 4b - 2 = 0$

This can be rearranged to find $b$ :

$4b = 4k^2 - 2$

$b = k^2 - rac{1}{2}$

Notice that for $b$ to be an integer, $k^2 - rac{1}{2}$ must also be an integer. However, since $rac{1}{2}$ is not an integer, $b$ cannot be an integer, leading us to the conclusion that:

$2b + 1$ would be odd, which contradicts our initial assumption that $b$ is an integer.

Step 3

Explain what can be deduced about the solutions of the equation.

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Answer

From our analysis, we have derived a contradiction when assuming that both $a$ and $b$ are integers while solving the equation $a^2 - 4b - 2 = 0$ . Therefore, we conclude that:

There are no solutions to the equation $a^2 - 4b - 2 = 0$ where $a$ and $b$ are integers.

Assume that a and b are integers such that a^2 - 4b - 2 = 0 9 (a) Prove that a is even - AQA - A-Level Maths Pure - Question 9 - 2022 - Paper 3

Worked Solution & Example Answer:Assume that a and b are integers such that a^2 - 4b - 2 = 0 9 (a) Prove that a is even - AQA - A-Level Maths Pure - Question 9 - 2022 - Paper 3

Prove that a is even.

Hence, prove that 2b + 1 is even and explain why this is a contradiction.

Explain what can be deduced about the solutions of the equation.

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