Question

applying factors to solve a problem
two positive, consecutive, odd integers have a product of 143.
complete the equation to represent finding x, the greater integer.
x(x - \boxed{}) = 143
what is the greater integer?
\boxed{}

Explanation:

Part 1: Completing the Equation

Step1: Analyze consecutive odd integers

Consecutive odd integers differ by 2. If \( x \) is the greater odd integer, the smaller one is \( x - 2 \).
So the equation is \( x(x - 2)=143 \).

Step2: Solve the equation \( x(x - 2)=143 \)

Expand the left side: \( x^{2}-2x = 143 \)
Rearrange to standard quadratic form: \( x^{2}-2x - 143 = 0 \)
Factor the quadratic: We need two numbers that multiply to -143 and add to -2. The numbers are -13 and 11. So \( (x - 13)(x + 11)=0 \)
Set each factor to zero: \( x - 13 = 0 \) or \( x + 11 = 0 \)
Solve for \( x \): \( x = 13 \) or \( x = -11 \). Since we want positive integers, we take \( x = 13 \).

Answer:

For the first blank, the answer is 2. For the greater integer, the answer is 13.