To use the Quadratic Formula, we substitute the values of into the expression on the right side of the formula.Then, we do all the math to simplify the expression.
So factoring out −2 will result in the common factor of (r – 3).
If we had gotten (−r 3) as a factor, then when setting that factor equal to zero and solving for r we would have gotten: There are many applications for quadratic equations.
We will see in the next example how using the Quadratic Formula to solve an equation with a perfect square also gives just one solution.
When we solved the quadratic equations in the previous examples, sometimes we got two solutions, sometimes one solution, sometimes no real solutions.
Is there a way to predict the number of solutions to a quadratic equation without actually solving the equation?
Yes, the quantity inside the radical of the Quadratic Formula makes it easy for us to determine the number of solutions.
Now, we will go through the steps of completing the square in general to solve a quadratic equation for .
It may be helpful to look at one of the examples at the end of the last section where we solved an equation of the form as you read through the algebraic steps below, so you see them with numbers as well as ‘in general.’ This last equation is the Quadratic Formula.
We cannot take the square root of a negative number.
So, when we substitute , , and into the Quadratic Formula, if the quantity inside the radical is negative, the quadratic equation has no real solution. The quadratic equations we have solved so far in this section were all written in standard form, .