Pearson Algebra 2 Common Core, 2011
PA
Pearson Algebra 2 Common Core, 2011 View details
7. The Quadratic Formula
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Exercise 9 Page 244

For a quadratic equation of the form the discriminant is given by the expression shown below.
The sign of the discriminant value tells us the number of real solutions the quadratic equation has. However, it cannot tell us by itself what the solutions are. We need to use the Quadratic Formula for this.
Notice that the is part of the formula.

Constructing a Counterexample

To give an counterexample to the exercise's statement, notice that we can have two different equations by swapping the values of and Since the discriminant depends on their product it would still be the same for both, but the solutions will not be the same. We can start by writing two equations in the standard form

Calculating the Discriminant

Let's calculate the discriminant for Equation I.
Substitute values and simplify
Notice that Equation II will have the same discriminant value, as is the same for both equations, and the product would be same as well.

Finding the Solutions Using the Quadratic Formula

Now we can use the Quadratic Formula for this case, to show that even though both equations have the same discriminant values as we said before, they will have different solutions. Recall that we know that the discriminant is in both cases.
Let's find the solutions for both equations by substituting the corresponding values into the above formula.
Equation (I) Equation (II)

We can see that the solutions are not the same. For Equation (I) the solutions are and while Equation (II) has and for solutions. Notice that there are infinitely many solutions satisfying the exercise's requirements. This is only an example solution.