Find two consecutive odd integers such that the square of the first integers increase by the second is 58

Answer 1

Let's call the first integer x.

Since the question states 2 consecutive odd integers, we will call the second integer x + 2.

"The square of the first integer increased by the second" translates to

x² + (x + 2)

Since the statement above equals to 58, we can set the above expression equal to 58.

x² + (x + 2) = 58

x² + x +2 = 58

Notice that this is a quadratic equation.

We can solve this by factoring, completing the square, or by using the quadratic formula.

Let's use the factoring method.

In order to factor, we must set the equation equal to 0.

Thus, we must subtract 58 to both sides of the equation and get

x² + x - 56 = 0

Now, we must find two integers that multiplies to -56 and adds to 1.

In this case, it would be -7 and 8.

Therefore, the trinomial factors into

(x - 7) · (x + 8)

Lastly, we must solve for the x values.

(x - 7) · (x + 8) = 0

x - 7 = 0 AND x + 8 = 0

x = 7 AND x = -8

Since the problem states the integers are odd, x must be 7.

Therefore, the two consecutive odd integers are 7 and 9.

*Please feel free to ask me any question for clarification.