Question

the product of two consecutive integers is 272. which quadratic equation can be used to find x, the smaller number?
\\( x^2 + 1 = 272 \\)
\\( x^2 - 1 = 272 \\)
\\( x^2 + x - 272 = 0 \\)
\\( x^2 + x + 272 = 0 \\)

Explanation:

Step1: Define consecutive integers

Let the smaller integer be \( x \), then the next consecutive integer is \( x + 1 \).

Step2: Set up the product equation

The product of the two consecutive integers is \( x(x + 1) \), and this is equal to 272. So we have the equation \( x(x + 1)=272 \).

Step3: Expand and rearrange the equation

Expanding \( x(x + 1) \) gives \( x^{2}+x \). So the equation becomes \( x^{2}+x = 272 \). Then, subtracting 272 from both sides to get the quadratic equation in standard form, we have \( x^{2}+x - 272 = 0 \).

Answer:

\( x^{2}+x - 272 = 0 \) (the third option: \( x^{2}+x - 272 = 0 \))