Final answer:
To solve the given math problem about consecutive integers, a system of equations was set up using the conditions provided. The quadratic equation was solved to determine the first integer, which led to the solution that the consecutive integers are 8, 9, and 10, corresponding to option D.
Step-by-step explanation:
The student is asking for help in solving a problem involving consecutive integers and their relationships using squares and products. To solve this, we will denote the first integer as x, thus making the other two consecutive integers x + 1 and x + 2. The problem states that the square of the first integer increased by the product of the other two equals 154, which gives us the equation:
x^2 + (x + 1)(x + 2) = 154
In order to find the solution, we must expand and simplify the equation, then solve for x:
- x^2 + x^2 + 2x + x + 2 = 154
- 2x^2 + 3x + 2 - 154 = 0
- 2x^2 + 3x - 152 = 0
Next, we find the factors of the quadratic equation that will sum up to 3 and multiply to give -304 (2 * -152). The factors are 19 and -16, so we rewrite the middle term using these factors and solve for x:
- 2x^2 + 19x - 16x - 152 = 0
- x(2x + 19) - 8(2x + 19) = 0
- (2x + 19)(x - 8) = 0
- x = -19/2 (not a valid solution for an integer problem) or x = 8
Thus, the consecutive integers are 8, 9, and 10.
From the options provided, the correct main answer is D) 8, 9, 10.