Question Video: Using Quadratic Equations to Solve Problems Mathematics

Video Transcript

A right triangle has sides of lengths 𝑛 centimeters, three multiplied by 𝑛 plus one centimeters, and three 𝑛 plus four centimeters. Find the length of its shortest side. We’re given a hint, which is “First, decide which side is the hypotenuse.”

In this problem, we have been given expressions for the lengths of the three sides of a right triangle all in terms of an unknown 𝑛. We know that the three sides in a right triangle are related to one another by the Pythagorean theorem. This tells us that in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse, which we can write as 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑎 and 𝑏 represent the lengths of the two shorter sides and 𝑐 represents the length of the hypotenuse.

We can use the Pythagorean theorem to form an equation in this unknown variable 𝑛. But first, as specified in the hint, we need to decide which side of the triangle is the hypotenuse.

The first side in the triangle has a length of 𝑛 centimeters. The next side has a length of three multiplied by 𝑛 plus one centimeters. Distributing the parentheses in this expression, this is equal to three 𝑛 plus three. And the final side of the triangle has a length of three 𝑛 plus four centimeters.

Now, as 𝑛 is the length of one of the sides in this triangle, it must be positive. So we know that 𝑛 is greater than zero. It follows then that 𝑛 must be less than three 𝑛 plus three because if we take a positive value, multiply it by another positive value, and then add a positive value, we’re going to get a larger number than we started with.

It’s also true, regardless of the value of 𝑛, that three 𝑛 plus four will have a greater value than three 𝑛 plus three. So we’ve ordered the expressions from smallest to largest. The shortest side of this triangle is the one represented by 𝑛, and the longest side of this triangle, which is also the hypotenuse, is represented by the expression three 𝑛 plus four. Now that we’ve determined which side of the triangle is the hypotenuse, we can substitute the three expressions into the Pythagorean theorem. This gives 𝑛 squared plus three 𝑛 plus three all squared is equal to three 𝑛 plus four all squared.

To simplify this equation, we need to distribute each set of parentheses. And when we do, we need to recall that when we square an expression, we are multiplying that expression by itself. So three 𝑛 plus three all squared is equal to three 𝑛 plus three multiplied by three 𝑛 plus three. Distributing these parentheses, perhaps using the FOIL method, gives nine 𝑛 squared plus nine 𝑛 plus nine 𝑛 plus nine, which simplifies to nine 𝑛 squared plus 18𝑛 plus nine.

In the same way, we expand three 𝑛 plus four all squared by multiplying three 𝑛 plus four by itself, which gives nine 𝑛 squared plus 24𝑛 plus 16. So our equation becomes 𝑛 squared plus nine 𝑛 squared plus 18𝑛 plus nine is equal to nine 𝑛 squared plus 24𝑛 plus 16.

We’re going to collect all of the terms on the same side of the equation, but we can see straightaway that we have nine 𝑛 squared on each side since these two terms will cancel one another out. We can then subtract 24𝑛 from each side of the equation, which gives 𝑛 squared minus six 𝑛 plus nine equals 16. And then we can subtract 16 from each side to give 𝑛 squared minus six 𝑛 minus seven equals zero. What we have is a quadratic equation in this unknown variable 𝑛.

Let’s see if we can solve this equation by factoring. We’re looking for two linear expressions which multiply together to give the quadratic. As the coefficient of 𝑛 squared is one, we know that the first term in each set of parentheses will be 𝑛 because 𝑛 multiplied by 𝑛 gives 𝑛 squared.

We’re then looking for two numbers to complete these parentheses, which have a specific set of properties. Firstly, the two numbers need to have a sum equal to the coefficient of 𝑛. That’s negative six. Secondly, they need to have a product equal to the constant term, which is negative seven. Now seven is a prime number. Its only factors are one and seven. We’re looking for a product of negative seven, though, which means we need one number to be negative and the other positive. If we choose the numbers to be negative seven and positive one, then these numbers have a product of negative seven. But they also have a sum of negative six.

We can use these two numbers to complete our parentheses then. If we wish, we could redistribute the parentheses, perhaps using the FOIL method, and confirm that we do indeed have 𝑛 squared minus six 𝑛 minus seven. So we factored correctly.

Next, we recall that if the product of two numbers or two expressions is equal to zero, then one of the individual expressions themselves must be equal to zero. This gives the two linear equations 𝑛 minus seven equals zero and 𝑛 plus one equals zero. Each of these equations can be solved in one step. To solve the equation 𝑛 minus seven equals zero, we add seven to each side, giving 𝑛 equals seven. And to solve the equation 𝑛 plus one equals zero, we subtract one from each side, giving 𝑛 equals negative one. We find then that there are two solutions to this quadratic equation: 𝑛 equals seven and 𝑛 equals negative one.

However, we should recall that 𝑛 represents the length of one of the sides in this right triangle, and so 𝑛 must be positive. This means that whilst 𝑛 equals negative one is a valid solution to this quadratic equation, it isn’t a possible value for 𝑛, and so our only solution is 𝑛 equals seven. We were asked to find the length of the shortest side of this triangle, which we decided was the side whose expression was 𝑛 centimeters. So our answer will be seven centimeters.

We should check this value though. And we can do this by evaluating the expressions for the other two sides and then confirming that these three values do indeed satisfy the Pythagorean theorem. If 𝑛 is equal to seven, then three 𝑛 plus three is equal to 21 plus three, which is 24. And three 𝑛 plus four is 21 plus four, which is 25.

We may recognize that seven, 24, 25 is a Pythagorean triple. But if not, we can use the Pythagorean theorem to check. Seven squared plus 24 squared is 49 plus 576, which is 625. 25 squared is also equal to 625. So this confirms that our answer is correct.

Using the Pythagorean theorem then and solving the resulting quadratic equation by factoring, we found that the length of the shortest side of this right triangle is seven centimeters.