Find the solution set of 𝑥 plus 17 over 𝑥 equals six in the set of real values, giving values to one decimal place.
In this question, it may not immediately be obvious how we can solve the equation. One strategy which will usually work is to try and eliminate any fractions first. If we multiply the second term 17 over 𝑥 by 𝑥, we know that the 𝑥’s will cancel. As a result, our start point will be to multiply both sides of our equation by 𝑥. Distributing the parentheses on the left-hand side, we know that 𝑥 multiplied by 𝑥 is 𝑥 squared. As already mentioned, multiplying 𝑥 by 17 over 𝑥 gives us 17𝑥 over 𝑥, which then simplifies to 17. The left-hand side of our equation is 𝑥 squared plus 17, and this is equal to six 𝑥.
Next, we can subtract six 𝑥 from both sides of our equation. This gives us 𝑥 squared minus six 𝑥 plus 17 equals zero. This is a quadratic equation written in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. One way of solving an equation of this type when 𝑎, 𝑏, and 𝑐 are constants and 𝑎 is nonzero is using the quadratic formula. This states that 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all divided by two 𝑎. In our equation, the coefficient of 𝑥 squared is equal to one. Therefore, 𝑎 is one. The coefficient of 𝑥 is negative six, so 𝑏 is equal to negative six. And the constant term 𝑐 is equal to 17. Substituting in these values, we have 𝑥 is equal to negative negative six plus or minus the square root of negative six squared minus four multiplied by one multiplied by 17 all divided by two multiplied by one.
At this stage, it is worth recalling a couple of key points. Firstly, for our equation to have real solutions, we know that the value of 𝑏 squared minus four 𝑎𝑐, known as the discriminant, must be greater than or equal to zero. This is because there are no real solutions to the square root of a negative number. It is also important to note that when we square a negative number on our calculator, we must put the number in brackets or parentheses. Simply typing negative six squared gives us an answer of negative 36. However, we know that when we square a negative number, we must have a positive answer. The square of negative six is 36 as we are multiplying negative six by itself.
Our equation simplifies to 𝑥 is equal to six plus or minus the square root of 36 minus 68 all divided by two. 36 minus 68 is equal to negative 32. As the square root of negative 32 has no real solutions, there will be no real solutions to this quadratic equation. This means that the solution is the empty set. There are no real solutions to the equation 𝑥 plus 17 over 𝑥 equals six.