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Steve and Alice realized that their ages are consecutive odd integers. The product of their ages is $195 .$ Steve is younger than Alice. Determine their ages by completing the square.

The 2 numbers are 13 and $15 .$

Algebra

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Section 1

Real Roots of a Quadratic Equation

Equations and Inequalities

Quadratic Functions

Complex Numbers

Polynomials

Campbell University

Oregon State University

Lectures

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The product of two consecu…

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The sum of the squares of …

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Okay. This question concerns the ages of two people. So Steve and Alice have ages that air Consecutive odd integers. So what does that mean? Well, consecutive odd integers like one and three more seven and nine are of the form to K minus one and two K plus one. So it says that the product of two of these odd consecutive energy rages are 1 95 So 1 95 equals to K minus one times two K plus one. And I'm not just using any intruders here because they have to be odd. That's why we have this two K minus one. Because that's the form of an odd number because two K is the form of uneven number. So we subtract one to get the form of a not number or doing this out. We get four k squared minus tu que plus two K minus one equals 1 95 and we can simplify this as four K squared minus one equals 1 95 or four K squared minus 1 96 equals zero. And actually, what we can do is pull out of four to get four times K squared minus 1 96 Divided by four. Wish we could do the math on this real fast. Cols And in 19 4 times with remainder three so 49 k squared minus 49 is equal to zero. And this four has nothing to do with our situation. So you can just cancel it because it's not gonna make cake with zero. It's not gonna make this expression equal to zero. Rather, so we know that K squared has to equal 49 and, well, okay can either be positive or negative seven. But we're dealing with positive numbers because we're dealing with age streaky. Cut off that K value. So now, looking back to the very beginning, we said that Steve is younger than Alice. So Steve has aged to K minus one, and Alice is older by two years. So her age is two K plus one. So if we plug in K equals seven for each of these, Steve is 14 minus one, which is 13 and Alice is 14 plus one, which is 15. And just like that, we've solved for each person's age

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