Using the quadratic formula, list all the roots of the equation ℎ of 𝑡 equals negative one-quarter 𝑡 squared plus three 𝑡 plus one. If necessary, round your answers to three decimal places.
We begin by recalling the definition of the roots of the equation. They are all the values of 𝑡 that satisfy the equation negative one-quarter 𝑡 squared plus three 𝑡 plus one equals zero. And usually, there are a number of ways we can achieve this. But we’re told to use the quadratic formula. This tells us that the solutions to the equation of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero are given by negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎.
Now, try not to worry about this plus or minus. This just means there are two separate calculations we need to do. We’ll need to work out the value of 𝑥 when we work out negative 𝑏 plus the square root all over two 𝑎 and then the other value of 𝑥, which is negative 𝑏 minus the square root all over two 𝑎. So let’s define what 𝑎, 𝑏, and 𝑐 are according to our formula. And we shouldn’t worry that the quadratic formula was in terms of 𝑥. This works for a quadratic equation in terms of 𝑡 too. 𝑎 is the coefficient of the 𝑡 squared here. It’s negative one-quarter. 𝑏 is equal to the coefficient of 𝑡; that’s three. And 𝑐 is the constant. Here, we’re going to let that 𝑏 equal to one.
Substituting into the quadratic formula, and we have 𝑡 is equal to negative three plus or minus the square root of three squared minus four times negative one-quarter times one all over two times negative one-quarter. Three squared, of course, is nine and four times negative one-quarter times one is negative one. So the discriminant, that’s the part inside the square root, becomes nine plus one, which is equal to 10. Similarly, two times negative one-quarter is negative one-half. So 𝑡 is equal to negative three plus or minus the square root of 10 all over negative one-half.
Now in fact, whilst we’re ready to type some numbers into our calculator, we could rewrite this slightly. Dividing by a half is the same as multiplying by two. So dividing by negative a half would be the same as multiplying by negative two. And we have negative three plus or minus the square root of 10 times negative two. Let’s look at the value of the first solution. That’s 𝑡 equals negative three plus the square root of 10 times negative two. That’s negative 0.32455. We’re going to round this to three decimal places.
So we look at the deciding digit; that’s the number five. And since this number is greater than or equal to five, we round up. And the first solution is negative 0.325. The second solution to our equation or the second root is negative three. minus the square root of 10 multiplied by negative two. This time that’s 12.32455 and so on. Once again, the deciding digit is five. And so we round up, and that gives us 12.325. And so the roots of the equation ℎ of 𝑡 equals negative one-quarter 𝑡 squared plus three 𝑡 plus one are 𝑡 equals negative 0.325 and 12.325.