Video Transcript
Find the solution set of the equation four plus three over 𝑥 equals 23 over 𝑥 squared, giving values to two decimal places.
The solution set of an equation is the set of all values of the unknown that satisfy the equation. This equation looks a little bit complex because it involves two terms where the unknown, 𝑥, is in the denominator of a fraction. We have 𝑥 in the denominator of the fraction here and 𝑥 squared in the denominator here. We can make this equation look a bit more familiar if we eliminate the denominators. To do this, we need to multiply both sides of the equation by the highest power of 𝑥 that appears in the denominators. So we need to multiply both sides by 𝑥 squared.
On the left-hand side, we have four multiplied by 𝑥 squared, which is four 𝑥 squared and then three over 𝑥 multiplied by 𝑥 squared, which is three 𝑥 squared over 𝑥. And this just simplifies to three 𝑥. On the right-hand side, the 𝑥 squareds cancel each other out, and we’re just left with 23. We can then subtract 23 from each side of the equation to give four 𝑥 squared plus three 𝑥 minus 23 is equal to zero. We should now recognize that this equation is in fact a quadratic equation in 𝑥. We can solve this equation by applying the quadratic formula.
The quadratic formula tells us that the roots of the general quadratic equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are real constants and 𝑎 is nonzero, are given by 𝑥 equals negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎. We can identify the values of 𝑎, 𝑏, and 𝑐 for this quadratic equation. The value of 𝑎 is the coefficient of 𝑥 squared, so it’s four. The value of 𝑏 is the coefficient of 𝑥, so it’s three. And the value of 𝑐 is the constant term; it’s equal to negative 23. We can now substitute these values of 𝑎, 𝑏, and 𝑐 into the quadratic formula to find the roots of our equation.
This gives 𝑥 equals negative three plus or minus the square root of three squared minus four times four times negative 23 all over two times four. This simplifies to negative three plus or minus the square root of nine minus negative 368 all over eight. And of course nine minus negative 368 is the same as nine plus 368, which is 377. So we have 𝑥 equals negative three plus or minus the square root of 377 all over eight.
The first root is found by taking negative three plus the square root of 377 all over eight. And evaluating this on a calculator gives 2.052. And the second root is found by taking negative three minus the square root of 377 all over eight, which evaluates to negative 2.802.
We’re asked to find the solution set of this equation, which means we need to write our answer as a set containing these two values. And we’re asked to round them to two decimal places. We can write the values in any order, but it’s perhaps more usual to write the smaller value first.
We found that the solution set of the equation four plus three over 𝑥 equals 23 over 𝑥 squared is the set containing the values negative 2.80 and 2.05, where each value is given to two decimal places.
