1. The Discriminant and Equations That Can Be Written in Quadratic Form
Section 10.3
The Discriminant and Equations That Can Be Written in Quadratic Form

2. Objectives
Use the discriminant to determine number and type of solutions
Solve equations that are quadratic in form
Solve application problems using quadratic equations

3. Objective 1: Use the Discriminant to Determine Number and Type of Solutions
The expression b2 – 4ac that appears under the radical symbol in the quadratic formula is called the discriminant. The discriminant can be used to predict what kind of solutions a quadratic equation has without solving it.
The Discriminant

4. EXAMPLE 1
Determine the number and type of solutions for each equation:
a. x2 + x + 1 = 0 b. 3x2 + 5x + 2 = 0
Strategy We will identify the values of a, b, and c in each equation. Then we will use those values to compute b2 – 4ac, the discriminant.
Why Once we know whether the discriminant is positive, 0, or negative, and whether it is a perfect square, we can determine the number and type of the solutions of the equation.

5. EXAMPLE 1 Solution a. For x2 + x + 1 = 0, the discriminant is:
Determine the number and type of solutions for each equation:
a. x2 + x + 1 = 0 b. 3x2 + 5x + 2 = 0
Solution
a. For x2 + x + 1 = 0, the discriminant is:
Since b2 – 4ac < 0, the solutions of x2 + x + 1 = 0 are two different imaginary numbers that are complex conjugates.
b. For 3x2 + 5x + 2 = 0, the discriminant is:
Since b2 – 4ac > 0, and b2 – 4ac is a perfect square, the solutions of 3x2 + 5x + 2 = 0 are two different rational numbers.

6. Objective 2: Solve Equations That Are Quadratic in Form
We have discussed four methods that are used to solve quadratic equations. The table below shows some advantages and disadvantages of each method.

7. Objective 2: Solve Equations That Are Quadratic in Form
To determine the most efficient method for a given equation, we can use the following strategy.
Strategy for Solving Quadratic Equations
See whether the equation is in a form such that the square root method is easily applied.
If step 1 does not apply, write the equation in ax2 + bx + c = 0 form.
See whether the equation can be solved using the factoring method.
If you can’t factor, solve the equation by the quadratic formula. Completing the square can also be used.
Equations that contain an expression, the same expression squared, and a constant term are said to be quadratic in form. One method used to solve such equations is to make a substitution.

8. EXAMPLE 2
Solve x4 – 3x2 – 4 = 0
Strategy Since the leading term, x4, is the square of the expression x2 in the middle term, we will substitute y for x2.
Why Our hope is that such a substitution will produce an equation that we can solve using one of the methods previously discussed.

9. EXAMPLE 2
Solve x4 – 3x2 – 4 = 0
Solution

10. Objective 3: Solve Application Problems Using Quadratic Equations

11. EXAMPLE 7
Household Appliances. The illustration shows a temperature control on a washing machine. When the warm setting is selected, both the hot and cold water pipes open to fill the tub in 2 minutes 15 seconds. When the cold setting is chosen, the tub fills 45 seconds faster than when the hot setting is used. How long does it take to fill the washing machine if the hot setting is used?
Analyze It is helpful to organize the facts of this shared-work problem in a table. We note that the hot and cold water inlets will be open for the same time: 2 minutes 15 seconds, or 135 seconds.
Assign Let x = the number of seconds it takes to fill the tub with hot water. Since the cold water inlet fills the tub in 45 seconds less time, x – 45 = the number of seconds it takes to fill the tub with cold water.

12. EXAMPLE 7
Household Appliances. …. How long does it take to fill the washing machine with if the hot setting is used?
Form
To determine the work completed by each inlet, multiply the rate by the time.
In shared-work problems, 1 represents one whole job completed. So we have

13. EXAMPLE 7
Household Appliances. …. How long does it take to fill the washing machine with if the hot setting is used?
Solve

14. EXAMPLE 7
Household Appliances. …. How long does it take to fill the washing machine with if the hot setting is used?
Solve
To solve this equation, we will use the quadratic formula, with a = 1, b = −315, and c = 6,075.

15. EXAMPLE 7
Household Appliances. The illustration shows a temperature control on a washing machine. When the warm setting is selected, both the hot and cold water pipes open to fill the tub in 2 minutes 15 seconds. When the cold setting is chosen, the tub fills 45 seconds faster than when the hot setting is used. How long does it take to fill the washing machine if the hot setting is used?
State We can discard the solution of 21 seconds, because this would imply that the cold water inlet fills the tub in a negative number of seconds (21 – 45 = –24). Therefore, the hot water inlet fills the washing machine tub in about 294 seconds, which is 4 minutes 54 seconds.
Check Use estimation to check the result.