Video Transcript
Find the solution set of the equation 𝑥 plus 21 over 𝑥 equals negative 10 in a set of real numbers.
In this question, we’re given this equation and asked to find the solution set. Remember that when we’re finding the solution set or solving, that simply means we’re finding a value or values of 𝑥. One way to begin solving this equation is to rearrange the equation so that we don’t have this denominator of 𝑥 in the term 21 over 𝑥. So we can multiply both sides of this equation by 𝑥.
On the left-hand side, distributing 𝑥 across the parentheses gives us 𝑥 squared, and then we have 𝑥 times 21 over 𝑥 or 21𝑥 over 𝑥, which simplifies to 21. On the right-hand side, negative 10 times 𝑥 is simply negative 10𝑥. Next, we can add 10𝑥 to both sides of the equation, which gives us 𝑥 squared plus 10𝑥 plus 21 is equal to zero. Notice that we’ve written it with the 𝑥 squared term and then the 𝑥-term and then the constant term equal to zero so that it’s easier to factor.
When we’re factoring a quadratic, we’ll have up to two sets of parentheses multiplied, giving us zero. We might notice that we have a coefficient of 𝑥 squared that’s simply one. So that means that both of these expressions in parentheses will have an 𝑥 at the start. Now, we need to find two terms which multiply to give 21 and add to give 10. Well, if we consider factor pairs of 21, we’ve got one and 21 or three and seven. However, when we add one and 21, we get 22 and not 10 which means that the factor pair for 21 will be three and seven, since they add to give us positive 10. That means that within our parentheses, we’ll have 𝑥 plus three and 𝑥 plus seven.
When we’re solving this, remember that if we have this value of 𝑥 plus three multiplied by this value of 𝑥 plus seven equals zero, then one of these must be zero. So 𝑥 plus three is equal to zero, or 𝑥 plus seven is equal to zero. When 𝑥 plus three is equal to zero, that means that 𝑥 must be equal to negative three. And when 𝑥 plus seven is zero, that means that 𝑥 must be equal to negative seven. Usually, it’s fine to give our answer in this form, but in this question we need to give the answer as a solution set. So our answer is the set containing negative three and negative seven.
