9k + 7 and 2k² + 3 are consecutive integers.
9k + 7 is the smaller integer.
Work out the value of the next consecutive integer.

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.