Solve Quadratic Equations by the Quadratic Formula
Goals • Solve quadratic equations using the
quadratic
formula.
Your Notes  VOCABULARY Quadratic formula A formula that gives the solutions of any quadratic equation in standard form 

THE QUADRATIC FORMULA The roots of the polynomial and the solutions of the quadratic equation
are
when 

Example 1 Multiple Choice Practice  
Which is one of the solutions to the
equation 

Solution  
Quadratic formula  
Substitute values in the quadratic formula: a = 2 , b = 3 , and c = 5 . 

Simplify.  
Simplify the square root. 

One solution is 1. The correct
answer is B 

Your Notes  Example 2 Use the quadratic formula  
Crabbing A crabbing net is thrown
from a bridge, which is 35 feet above the water. If the net's initial velocity is 10 feet per second, how long will it take the net to hit the water? Solution The net's initial velocity is v = 10 feet per second and the net's initial height is s = 35 feet. The net will hit the water when the height is 0 feet. 

Vertical motion model  
Substitute for h, v, and s.  
Substitute values in the quadratic formula: a = 16 , b= 10 , and c = 35 . 

Simplify.  
The solutions are and  
Because time cannot be a negative number, disregard the negative solution. 
So, the net
will hit the water in about 1.82 seconds. 

Guided Practice Complete the following exercises.  
1. Use the quadratic formula to solve
and 

2. In Example 2, suppose the net was
thrown with an initial velocity of 5 feet per second from a height of 20 feet. How long would it take the net to hit the water? about 1.29 seconds 

Your Notes  METHODS FOR SOLVING QUADRATIC EQUATIONS  
Methods  When to Use  
Factoring  Use when a quadratic equation can be factored easily. 

Graphing  Use when approximate solutions are adequate. 

Finding  Use when solving an equation that can  
square roots  be written in the form x^{2} = d .  
Completing  Can be used for any quadratic equation  
the square  Can be used for any quadratic equation the square but is simplest to apply when a = 1 and b is an even number. 

Quadratic  Can be used for any quadratic equation.  
formula  
Example 3 Choose a solution method  
Tell what method(s) you would use to
solve the quadratic equation. Explain your choice(s). Solution a. The quadratic equation cannot be factored easily and completing the square would result in many fractions . So, the equation can be solved using the quadratic formula . b. The quadratic equation can be solved using square roots because the equation can be written in the form x^{2} = d. 

Guided Practice Complete the following exercise.  
Homework  3. Tell what method(s) you would use
to solve . Explain your choices(s). factoring: the expression can be factored easily; completing the square: the equation is of the form where a= 1 and b is an even number. 