Quadratic Equations and Applications Objective: To use different methods to solve quadratic equations.

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Presentation transcript:

Quadratic Equations and Applications Objective: To use different methods to solve quadratic equations.

Factoring Factoring is an effective way to solve quadratic equations.

Factoring Solve the following:

Factoring Solve the following: This is the easiest type because you always start this way.

Factoring Solve the following: This is the easiest type because you always start this way. Since the last term is negative, the signs must be opposite.

Factoring Solve the following: Since the middle term is – 5x, that means that the outer and inner terms are different by 5, and the larger term is negative.

Factoring Solve the following: Since the middle term is – 5x, that means that the outer and inner terms are different by 5, and the larger term is negative. We now need factors of 24 that are different by 5.

Factoring Solve the following: Since the middle term is – 5x, that means that the outer and inner terms are different by 5, and the larger term is negative. We now need factors of 24 that are different by 5.

Factoring Solve the following: We know that if two things multiplied together equal zero, one of them must be zero. or

Factoring Solve the following:

Factoring Solve the following: Since the last term is positive, both factors are the same sign, and since the middle term is positive, they are both positive.

Factoring Solve the following: Since the last term is positive, both factors are the same sign, and since the middle term is positive, they are both positive. We need factors of 24 that add to get 10.

Factoring Solve the following: Since the last term is positive, both factors are the same sign, and since the middle term is positive, they are both positive. We need factors of 24 that add to get 10.

Factoring Solve the following: Solve: or

You Try Solve the following:

You Try Solve the following: Since the last term is positive, both factors are the same sign, and now since the middle term is negative, they are both negative. We need factors of 36 that add to get 13.

Factoring Solve:

Factoring Solve: First, set the equation equal to zero.

Factoring Solve: First, set the equation equal to zero. This is how we start:

Factoring Solve: First, set the equation equal to zero. This is how we start: We need factors of 4 that will make the outer and inner terms add to get 9.

Factoring Solve: First, set the equation equal to zero. This is how we start: We need factors of 4 that will make the outer and inner terms add to get 9.

Factoring Solve:

Factoring Solve: With only two terms, we look for a greatest common factor. or

Square Roots Solve:

Square Roots Solve: First, isolate the variable and get the x 2 alone.

Square Roots Solve: First, isolate the variable and get the x 2 alone. Square root both sides.

Square Roots Solve:

Square Roots Solve: The x is by itself as much as it can be, so square root both sides.

Square Roots Solve: The x is by itself as much as it can be, so square root both sides. Add 3 to both sides to solve for x.

Square Roots You Try:

Square Roots You Try:

Class Work Page even 22, 24, 26, 28

Homework Page odd odd

Completing the Square When completing the square, we follow the same technique each time.

Completing the Square When completing the square, we follow the same technique each time. 1.Variables on one side, constants on the other.

Completing the Square When completing the square, we follow the same technique each time. 1.Variables on one side, constants on the other. 2.Make sure the x 2 is a 1x 2.

Completing the Square When completing the square, we follow the same technique each time. 1.Variables on one side, constants on the other. 2.Make sure the x 2 is a 1x 2. 3.Take half of the middle term and square it.

Completing the Square When completing the square, we follow the same technique each time. 1.Variables on one side, constants on the other. 2.Make sure the x 2 is a 1x 2. 3.Take half of the middle term and square it. 4.Add that number to both sides.

Completing the Square When completing the square, we follow the same technique each time. 1.Variables on one side, constants on the other. 2.Make sure the x 2 is a 1x 2. 3.Take half of the middle term and square it. 4.Add that number to both sides. 5.Solve.

Completing the Square Solve:

Completing the Square Solve: Add eight to both sides.

Completing the Square Solve: Add eight to both sides. Take half of 2, square it, and add it to both sides.

Completing the Square Solve: Add eight to both sides. Take half of 2, square it, and add it to both sides. Solve.

Completing the Square Solve: Add eight to both sides. Take half of 2, square it, and add it to both sides. Solve.

Completing the Square You Try:

Completing the Square Solve:

Completing the Square Solve:

Completing the Square Solve:

Completing the Square You Try:

Completing the Square Solve:

Completing the Square Solve:

Completing the Square Solve:

Completing the Square Solve:

Completing the Square Solve:

Completing the Square Solve:

Completing the Square Solve:

Rationalize Rationalize the following :

Rationalize Rationalize the following:

Rationalize Rationalize the following:

Rationalize Rationalize the following :

Rationalize You Try.

Rationalize You Try.

The Quadratic Formula Given a quadratic equation of the form: We can solve it using the quadratic formula:

The Quadratic Formula Solve using the quadratic formula.

The Quadratic Formula Solve using the quadratic formula. Set the equation equal to zero.

The Quadratic Formula Solve using the quadratic formula. Set the equation equal to zero. a=1, b=3, c=-9

The Quadratic Formula Solve using the quadratic formula. Set the equation equal to zero. a=1, b=3, c=-9 Use the quadratic equation.

The Quadratic Formula Solve using the quadratic formula. Set the equation equal to zero. a=1, b=3, c=-9 Use the quadratic equation.

The Quadratic Formula Solve using the quadratic formula. Set the equation equal to zero. a=1, b=3, c=-9 Use the quadratic equation.

The Quadratic Formula Solve using the quadratic formula.

The Quadratic Formula Solve using the quadratic formula. Set the equation equal to zero. a=1, b=-6, c=-4

The Quadratic Formula Solve using the quadratic formula. Set the equation equal to zero. a=1, b=-6, c=-4 Use the quadratic equation.

Class Work Page , 37, 38

Class Work Page , 37, 38 68, 72, 74, 76

Example 7 A bedroom is 3 feet longer than it is wide and has an area of 154 square feet. Find the dimensions of the room.

Example 7 A bedroom is 3 feet longer than it is wide and has an area of 154 square feet. Find the dimensions of the room. The width will be x The length will be x + 3

Example 7 A bedroom is 3 feet longer than it is wide and has an area of 154 square feet. Find the dimensions of the room. The width will be x The length will be x + 3 Since the area is length x width, the equation is: x(x+3) = 154

Ex. 7 We need to solve the equation to find x.

Ex. 7 We need to solve the equation to find x.

Ex. 7 Since x is the width of a room, it can’t be negative. The only answer we will use is x = 11. This means the width is 11 and the length is 14.

Example 9 From 2000 to 2007, the estimated number of Internet users I (in millions) in the United States can be modeled by the quadratic equation where t represents the year, with t = 0 being In what year will the number of Internet users surpass 180 million?

Ex. 9 We will use our calculator to solve the following problem.

Ex. 9 We will use our calculator to solve the following problem.

Ex. 9 We will use our calculator to solve the following problem.

Example 10 An L-shaped sidewalk from the athletic center to the library on a college campus is shown. The length of one sidewalk forming the L is twice as long as the other. The length of the diagonal between the two buildings is 32 feet. How many feet does a person save by walking on the diagonal?

Example 10 An L-shaped sidewalk from the athletic center to the library on a college campus is shown. The length of one sidewalk forming the L is twice as long as the other. The length of the diagonal between the two buildings is 32 feet. How many feet does a person save by walking on the diagonal? Let x = one path Let 2x = the other path

Example 10 An L-shaped sidewalk from the athletic center to the library on a college campus is shown. The length of one sidewalk forming the L is twice as long as the other. The length of the diagonal between the two buildings is 32 feet. How many feet does a person save by walking on the diagonal? Let x = one path Let 2x = the other path We will use the Pythagorean Theorem to solve.

Ex. 10

Homework Page mult. of odd 109,124

Lesson Quiz Solve by completing the square. 1.

Lesson Quiz Solve by completing the square. 2.

Lesson Quiz Solve by using the quadratic formula. 3.

Lesson Quiz Solve by using the quadratic formula 4.