Determine the type of the roots of the equation 𝑥 plus 36 over 𝑥 is equal to 12.
In order to answer this question, we firstly need to rewrite our equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero. We can begin to do this by multiplying both sides of the equation by 𝑥. 𝑥 multiplied by 𝑥 is equal to 𝑥 squared. Multiplying 36 over 𝑥 by 𝑥 gives us 36. On the right-hand side, multiplying 12 by 𝑥 gives us 12𝑥. We can then subtract 12𝑥 from both sides, giving us 𝑥 squared minus 12𝑥 plus 36 is equal to zero.
At this stage, we have two possible ways to proceed. Firstly, we could just try and solve the equation. One way of doing this is by factoring. 𝑥 squared minus 12𝑥 plus 36 is equal to 𝑥 minus six multiplied by 𝑥 minus six. For this expression to be equal to zero, then one of the factors must equal zero, either 𝑥 minus six equals zero or 𝑥 minus six equals zero. Adding six to both sides of these equations, we see that both solutions are 𝑥 is equal to six. This means that the roots of our equation are real and equal.
An alternative method to work at the type of roots of a quadratic equation is to consider the discriminant. This is the value of 𝑏 squared minus four 𝑎𝑐. In this question, 𝑎 is equal to one. It is the coefficient of 𝑥 squared. The coefficient of 𝑥, 𝑏, is equal to negative 12, and the constant term 𝑐 is equal to 36. Substituting in these values, we have negative 12 squared minus four multiplied by one multiplied by 36.
Squaring a negative number gives a positive answer. Therefore, negative 12 squared is equal to 144. Four multiplied by one multiplied by 36 is also equal to 144. 144 minus 144 is equal to zero. If the discriminant 𝑏 squared minus four 𝑎𝑐 is equal to zero, we know that the roots of the quadratic equation are real and equal. This confirms the answer found using our first method.
The roots of the equation 𝑥 plus 36 over 𝑥 is equal to 12 are real and equal.