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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
a ≠ 0 and

  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 x2 = 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 x2 = 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.