MTH 065 Class Notes Lecture 16 (4.1 and 4.2) Lesson 4.1: Solving Quadratic Equations: Graphing and Square Root Methods There are several methods we can use to solve a quadratic equation.
Using Graphs to Solve Quadratic Equations Solving by the Graphing Method To solve the quadratic equation Graph Find the x-intercept(s)
:
The solutions to the equation are the -coordinates of the x-intercepts. The solutions can also be called zeros of the quadratic function because they are values that make the equation equal zero. The number of solutions obtained from a quadratic equation depends upon the position of the parabola relative to the horizontal axis. This leaves three possibilities: When the parabola doesn’t touch the x-axis, it touches in a single spot, or intersects the x-axis at twice.
No solutions
One solution: (x = 1.5)
Two solutions: (x = -1, 2)
It is unlikely that the graph of every parabola will intersect the x-axis at an easy-to-read spot. More frequently, we will need to use our calculators to determine where the graph intersects the x-axis. In order to find the intersection between the parabola and the x-axis we need to know the line which represents the x-axis, that is y = 0. In our previous Modules, we used the CALC feature on our TI-83/84, this time we can use 2:Zero to find where our parabola intersects with y = 0.
Practice 1 Solve the equation
by graphing. Round answers to 2 decimal places.
Solutions: x = 1.78 or x = 0.28 Standard Form of a Quadratic Equation A quadratic equation written in the form , is said to be in standard form. To solve a quadratic equation by graphing, we will first write the equation in standard form. Note: Just remember that if the parabola does not intersect with the x-axis, then there will be no real solution to the quadratic equation. Practice 2 Solve the following equations by graphing. Round your answers to 2 decimal places. (a)
No real solutions (b)
(c)
Using the Square Root Method to Solve Quadratic Equations Solving by the Square Root Method If a quadratic equation fits the form , then it can be solved by the square root method.
Rewrite the equation so it is in the form Then √ and √ . This can be written as
Don’t forget the
√ .
. If you leave it off, you only have one of the answers.
Note: If you receive an answer with a negative under a square root, then there are no real solutions. Practice 3 Solve the following equations by the square root method. Express your answers both in exact form and as decimals rounded to 2 decimal places. (a) √ √ √ (b)
√
√ √ There are no real solutions
(c)
√
√ √
(d) (
)
√
√ √
Lesson 4.2: Solving Quadratic Equations: Quadratic Formula From Section 4.1, we found that we could find the solutions to a quadratic equation by both graphing as well as the square root method. The solutions from the graph are the x-intercepts, that is where our quadratic equation is equal to zero or: Mathematicians have taken this equation and solved it for x, creating what is now called the quadratic formula. Solving by the Quadratic Formula For a quadratic equation in standard form, , the solutions to the equation are found by the quadratic formula: √
When we have a quadratic equation and use the quadratic formula, every place we substituted in a value, we put it in parenthesis. You’re less likely to make a sign mistake that way. In addition, once we have solved the square root in the problem, we break out the two solutions, one for adding the terms in the numerator and one for subtracting them. Practice 1 A. Use the quadratic formula to find exact solutions. Check your answers. (a) We must first put this equation into standard form by moving every term to one side, this will set one side of the equation to 0.
a = 2, b = -5, c = 3 (
)
√(
) ( )
(
)
√ ( )
( )( )
√ √
OR OR OR Check: ( (
)
(
)
)
( )( )
( (
) )
(
)
( )
( )
B. a = 1, b = 7, c = 4 ( )
( )( ) √( ) ( ) ( )( ) √ ( )
( ) √
√ √
OR
√
C. a = -2, b = 3, c = -1 ( )
( )( ) √( ) ( ) ( )( ) √ ( )
( ) √ √
OR OR OR Note: The quadratic formula will work regardless of the type of solutions. For example, rational numbers as well as integers.
Practice 2 Use the quadratic formula to solve the following quadratic equations. Round your answers to 3 decimal places. Check your answers. (a) a = 3, b = -4, c = 6 (
)
√(
) ( )
(
)
√ ( )
( )( ) ( )( )
√ √ Since we have a negative value in the square root, there are no real solutions. (b) a = 16, b = -8, c = 1 (
)
)
√( (
(
)
√ (
(
)( )
) (
)( )
)
√ √
As you’ve noticed from these examples, the number under the square root symbol, the radicand, is the key to the number of solutions. The radicand from the quadratic formula, , is so important that it has a special name, the discriminant. Here’s how we can use the discriminant. If the discriminant,
Discriminant Test , of the quadratic formula is:
positive, there are two real solutions zero, there is one distinct real solution negative, there are no real solutions
Note: We must first have the equation into standard form.
Practice 3 For each quadratic equation below, use the discriminant to determine the number of solutions. If there are real solutions, use the quadratic formula to solve the equation. Round to 3 decimal places where necessary. (a)
a = 3, b = -2, c = 4 (
)
( )( )
( )( )
No real solutions (b)
a= (
)
(
)(
)
(
(
) (
√ )
)(
)
, b = -4, c = -8
(c)
a = , b = -4, c = -7 (
)
( )(
)
( (
) √ ( ) ) ( )
OR OR OR
( )(
)
