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AQA GCSE · Question 24.1 · Algebra
9k + 7 and 2k² + 3 are consecutive integers.
9k + 7 is the smaller integer.
Work out the value of the next consecutive integer.
9k + 7 and 2k² + 3 are consecutive integers.
9k + 7 is the smaller integer.
Work out the value of the next consecutive integer.
How to approach this question
1. "Consecutive integers" means one number comes directly after the other (e.g., 7 and 8).
2. If 9k + 7 is the smaller integer, then the larger integer, 2k² + 3, must be exactly one more than it.
3. Set up an equation: (9k + 7) + 1 = 2k² + 3.
4. Simplify and rearrange this equation into the standard quadratic form: ax² + bx + c = 0.
5. Solve the quadratic equation for k. You will likely get two possible values for k.
6. The question asks for the value of the *next consecutive integer*, which is the larger of the two original expressions. Substitute your valid value(s) of k into the expression 2k² + 3 to find the answer.
Full Answer
Since they are consecutive integers and 9k + 7 is the smaller one, the next integer is one greater.
So, (9k + 7) + 1 = 2k² + 3
9k + 8 = 2k² + 3
Rearrange this into a standard quadratic equation (ax² + bx + c = 0).
0 = 2k² - 9k + 3 - 8
2k² - 9k - 5 = 0
Now we need to solve this quadratic equation. We can try to factorise it. We need two numbers that multiply to give 2 * -5 = -10 and add to give -9. These numbers are -10 and +1.
Rewrite the middle term:
2k² - 10k + 1k - 5 = 0
Factorise by grouping:
2k(k - 5) + 1(k - 5) = 0
(2k + 1)(k - 5) = 0
This gives two possible solutions for k:
2k + 1 = 0 => k = -1/2
k - 5 = 0 => k = 5
Let's test both values.
If k = 5:
Smaller integer = 9(5) + 7 = 45 + 7 = 52
Larger integer = 2(5)² + 3 = 2(25) + 3 = 50 + 3 = 53
52 and 53 are consecutive integers. This solution works.
If k = -1/2:
Smaller integer = 9(-1/2) + 7 = -4.5 + 7 = 2.5
Larger integer = 2(-1/2)² + 3 = 2(1/4) + 3 = 0.5 + 3 = 3.5
2.5 and 3.5 are consecutive integers. This solution also works.
The question asks for the value of the *next* consecutive integer, which is the larger one (2k² + 3).
If k=5, the next integer is 53.
If k=-1/2, the next integer is 3.5. Since the question refers to "integers", k=5 is the intended solution.
The question asks for the value of the next consecutive integer. This is the value of 2k² + 3.
Using k=5, the value is 53.
Answer: 53
The problem states that 9k + 7 and 2k² + 3 are consecutive integers, with 9k + 7 being the smaller one. This means that if we add 1 to the smaller integer, we get the larger integer.
So, we can form the equation:
(9k + 7) + 1 = 2k² + 3
9k + 8 = 2k² + 3
Now, we rearrange this into a standard quadratic equation of the form ax² + bx + c = 0:
0 = 2k² - 9k + 3 - 8
2k² - 9k - 5 = 0
We can solve this quadratic equation by factorising. We are looking for two numbers that multiply to give (2 × -5) = -10 and add to give -9. The numbers are -10 and +1.
We split the middle term:
2k² - 10k + k - 5 = 0
Factorise in pairs:
2k(k - 5) + 1(k - 5) = 0
(2k + 1)(k - 5) = 0
This gives two possible solutions for k:
1. 2k + 1 = 0 => 2k = -1 => k = -1/2
2. k - 5 = 0 => k = 5
Since the question is about consecutive *integers*, the expressions 9k+7 and 2k²+3 must evaluate to integers.
If k = -1/2, the first "integer" is 9(-1/2) + 7 = -4.5 + 7 = 2.5, which is not an integer. So we discard this solution.
If k = 5, the first integer is 9(5) + 7 = 45 + 7 = 52. This is an integer.
The second integer is 2(5)² + 3 = 2(25) + 3 = 50 + 3 = 53. This is also an integer.
The two integers are 52 and 53, which are consecutive.
The question asks for the value of the *next* consecutive integer, which is the larger one.
The value is 53.
Common mistakes
✗ Setting up the initial equation incorrectly.
✗ Errors when rearranging into the quadratic form.
✗ Errors when factorising or solving the quadratic equation.
✗ Finding the value of k but not using it to find the value of the integer, which is what the question asks for.
✗ Not checking both solutions for k to see if they produce integers.
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