Find the solution set of the equation 𝑥 squared minus three 𝑥 plus one equals zero, giving values correct to two decimal places.
In order to find the solutions to this quadratic equation, we will use the quadratic formula. This states that, for any quadratic equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 is nonzero, then 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all divided by two 𝑎. It is important to note that we will only have real solutions if 𝑏 squared minus four 𝑎𝑐, known as the discriminant, is greater than or equal to zero.
The equation in this question is already written in the correct form. The coefficient of 𝑥 squared is one. Therefore, 𝑎 equals one. 𝑏 is the coefficient of 𝑥, in this question negative three, and 𝑐 is the constant equal to one. Substituting these values into the quadratic formula, we have 𝑥 is equal to negative negative three plus or minus the square root of negative three squared minus four multiplied by one multiplied by one all divided by two multiplied by one. This simplifies to three plus or minus the square root of nine minus four all divided by two, which in turn is equal to three plus or minus the square root of five all divided by two.
We have two solutions to the quadratic equation: either 𝑥 is equal to three plus root five divided by two or 𝑥 is equal to three minus root five all divided by two. Typing both of these into the calculator, we get 𝑥 is equal to 2.6180 and so on and 𝑥 is equal to 0.3819 and so on.
We are asked to round these to two decimal places. The solution set of the equation 𝑥 squared minus three 𝑥 plus one equals zero contains the values 2.62 and 0.38 correct to two decimal places.