2. Equations and Inequalities

Use the Discriminant to Determine the Number and Type of Solutions

Quadratic Equations

IB Sample Exam Question, Paper 1 Q9

Solving for 'X' ;  Quadratic Equation / Pythagorean Theorem - Example 1

Solve each equation for the specified variable. (Assume no denominators are 0.) See Example 8. r = r_0+(1/2)at^2, for t

In Exercises 35–46, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.

x^2 - 10x

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9. 4x^2 = -6x + 3

Solve each equation in Exercises 83–108 by the method of your choice.

2x^2 - x = 1

Word Problems Using the Pythagorean Theorem - Example 2

In Exercises 71–72, without solving the given quadratic equation, determine the number and type of solutions. 9x^2 = 2-3x

Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer, then x+1 represents the next consecutive integer.If x represents an even integer, then x+2 represents the next consecutive even integer.If x represents an odd integer, then x+2 represents the next consecutive odd integer. The difference of the squares of two positive consecutive odd integers is 32. Find the integers.

Solving Trigonometric Equations Using the Quadratic Formula - Example 2

Solve each equation in Exercises 83–108 by the method of your choice.

3x^2 = 60

In Exercises 35–46, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.

x^2 - (2/3)x

Solve each equation in Exercises 15–34 by the square root property.

3x^2 = 27

Solve each equation in Exercises 15–34 by the square root property.

3(x - 4)^2 = 15

Solve each equation using completing the square. See Examples 3 and 4. -2x^2 + 4x + 3 = 0

Solve each equation using the square root property. See Example 2. (4x + 1)^2 = 20

In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation.

x^2 - 2x + 1 = 0

Solve each equation in Exercises 15–34 by the square root property.

(3x + 2)^2 = 9

In Exercises 115–122, find all values of x satisfying the given conditions.

y1 = x - 1, y2 = x + 4 and y1y2 = 14

For each equation, (b) solve for y in terms of x. See Example 8. 2x^2 + 4xy - 3y^2 = 2

In Exercises 115–122, find all values of x satisfying the given conditions.

y = 2x^2 - 3x and y = 2

In Exercises 35–46, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.

x^2 - 7x

Solving Word Problems in Distance, Rate, and Time Using Quadratics - Example 2

Word Problems Using the Pythagorean Theorem - Example 3

Solve each problem. See Examples 1. Dimensions of a Parking Lot. A parking lot has a rectangular area of 40,000 yd2. The length is 200 yd more than twice the width. Find the dimensions of the lot.

Solve each equation in Exercises 83–108 by the method of your choice.

x^2 - 6x + 13 = 0

Solve each equation in Exercises 83–108 by the method of your choice.

x^2 - 2x = 1

Solving a Quadratic Equation: Factoring a Simple Example

Using the Discriminant for Quadratic Equations

Use the Quadratic Formula

Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer, then x+1 represents the next consecutive integer.If x represents an even integer, then x+2 represents the next consecutive even integer.If x represents an odd integer, then x+2 represents the next consecutive odd integer. The sum of the squares of two consecutive odd integers is 202. Find the integers

Solve each equation in Exercises 47–64 by completing the square.

3x^2 - 2x - 2 = 0

Solve each equation in Exercises 65–74 using the quadratic formula.

x^2 + 8x + 15 = 0

When the sum of 1 and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number.

Factoring Trinomials by Trial and Error - Ex 2

Solve each equation in Exercises 47–64 by completing the square.

x^2 - 6x - 11 = 0

Quadratic Equations - Completing the Square

Solve each equation in Exercises 15–34 by the square root property.

3x^2 - 1 = 47

Solve each equation using the quadratic formula. See Examples 5 and 6. -4x^2 = -12x + 11

Solve each equation for the specified variable. (Assume no denominators are 0.) See Example 8. F = kMv^2/r , for v

Solve each equation in Exercises 83–108 by the method of your choice.

1/x + 1/(x + 3) = 1/4

 Definition of A Quadratic Equation In One Variable

Algebra II: Quadratic Equations (Level 3 of 3)

Solving for 'X' ;  Quadratic Equations Involving the Pythagorean Theorem - Example 3

Algebra II: Quadratic Equations (Level 2 of 3)

Solve each equation in Exercises 47–64 by completing the square.

2x^2 - 7x + 3 = 0

Factoring Trinomials: Factor by Grouping - ex 1

Solve Quadratic Equations by Factoring a Trinomial / Example 2.8

Using the Discriminant to Predict the Types of Solutions to a Quadratic Equation - Example 2

Use Choices A–D to answer each question. A. 3x^2 - 17x - 6 = 0 B. (2x + 5)^2 = 7 C. x^2 + x = 12 D. (3x - 1)(x - 7) = 0
Only one of the equations is set up so that the values of a, b, and c can be determined immediately. Which one is it? Solve it.

Solving Quadratic Equations using the Quadratic Formula - Example 3

Solve each equation using the zero-factor property. See Example 1. x^2 - 5x + 6 = 0

Completing the Square Example 2 Solve Quad. Equations

Radius of a CanA can of Blue Runner Red Kidney Beans has surface area 371 cm^2. Its height is 12 cm. What is the radius of the circular top? Round to the nearest hun-dredth.

Solve each equation for the specified variable. (Assume no denominators are 0.) See Example 8. s = (1/2)gt^2, for t

Solve each equation in Exercises 83–108 by the method of your choice.

(2x - 5)(x + 1) = 2

Solve each equation in Exercises 83–108 by the method of your choice.

x^2 - 4x + 29 = 0

In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation.

2x^2 - 11x + 3 = 0

Solve each equation using the zero-factor property. See Example 1. x^2 - 64 = 0

 Solve Quadratic Equations by Factoring a Trinomial / Example 2.7

Algebra II: Quadratic Equations - Factoring (Level 6 of 10)

Solving Quadratic Equations - Factoring and Using the Quadratic Formula

Solve each equation in Exercises 65–74 using the quadratic formula.

x^2 - 6x + 10 = 0

Solve each equation using the zero-factor property. See Example 1. x^2 + 2x - 8 = 0

Solve each equation in Exercises 1 - 14 by factoring.

6x^2 + 11x - 10 = 0

Completing the Square to Solve Quadratic Equations: More Examples - 4

Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer, then x+1 represents the next consecutive integer.If x represents an even integer, then x+2 represents the next consecutive even integer.If x represents an odd integer, then x+2 represents the next consecutive odd integer. Find two consecutive odd integers whose product is 63.

Solving Trigonometric Equations Using the Quadratic Formula - Example 1

Using the Discriminant to Predict the Types of Solutions to a Quadratic Equation - Example 3

Solve each equation using the quadratic formula. See Examples 5 and 6. x^2 = 2x - 5

Solve each equation in Exercises 83–108 by the method of your choice.

3/(x - 3) + 5/(x - 4) = (x^2 - 20)/(x^2 - 7x + 12)

Which equation has two real, distinct solutions? Do not actually solve.
A. (3x-4)² = -9
B. (4-7x)² = 0
C. (5x-9)(5x-9) = 0
D. (7x+4)² = 11

Solve each equation in Exercises 15–34 by the square root property.

(x + 3)^2 = - 16

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9. 3x^2 + 5x + 2 = 0

Solve each equation using the quadratic formula. See Examples 5 and 6. 2/3x^2 + 1/4x = 3

Match the equation in Column I with its solution(s) in Column II. x^2 = -25

Using the Quadratic Formula

Solving Quadratic Equations using the Quadratic Formula - Example 1

Solving a Geometry Word Problem by Using Quadratic Equations - Example 1

Using the Discriminant to Predict the Types of Solutions to a Quadratic Equation - Example 1

Completing the Square to Solve Quadratic Equations: More Examples - 1

Solve each equation in Exercises 1 - 14 by factoring.

3x^2 - 2x = 8

Derivation of the Quadratic Formula

Solve each equation in Exercises 1 - 14 by factoring.

2x(x - 3) = 5x^2 - 7x

Solve each equation in Exercises 15–34 by the square root property.

5x^2 = 45

Solving Quadratic Equations by Factoring - Basic Examples

 Using the Quadratic Formula

Solve each equation in Exercises 15–34 by the square root property.

(4x - 1)^2 = 16

Create and solve quadratic equations

Solve each equation in Exercises 47–64 by completing the square.

3x^2 - 5x - 10 = 0

Solve each equation using completing the square. See Examples 3 and 4. 3x^2 + 2x = 5

Algebra II: Quadratic Equations - Factoring (Level 9 of 10)

Solve each equation in Exercises 83–108 by the method of your choice.

5x^2 + 2 = 11x

The Zero-Product Property

Match the equation in Column I with its solution(s) in Column II. x^2 - 5 = 0

Solve each equation using the quadratic formula. See Examples 5 and 6. 1/2x^2 + 1/4x - 3 = 0

Solve each equation in Exercises 1 - 14 by factoring.

10x - 1 = (2x + 1)^2

Solve each cubic equation using factoring and the quadratic formula. See Example 7. 
x^3 - 27 = 0

In Exercises 123–124, list all numbers that must be excluded from the domain of each rational expression.

3/(2x^2 + 4x - 9)

Use Choices A–D to answer each question. A. 3x^2 - 17x - 6 = 0 B. (2x + 5)^2 = 7 C. x^2 + x = 12 D. (3x - 1)(x - 7) = 0 
Which equation is set up for direct use of the square root property? Solve it

Completing the Square to Solve Quadratic Equations: More Examples - 3

Solve each equation in Exercises 1 - 14 by factoring.

x^2 = 8x - 15

Solve each equation in Exercises 47–64 by completing the square.

x^2 + 4x + 1 = 0

Solve each equation in Exercises 83–108 by the method of your choice.

2x^2 - 7x = 0

Solve each equation in Exercises 83–108 by the method of your choice.

x^2 = 4x - 7

(Modeling)Solve each problem. See Example 3.Height of a ProjectileA projectile is launched from ground level with an initial velocity of v_0 feet per second. Neglecting air resistance, its height in feet t seconds after launch is given by s=-16t^2+v_0t. In each exercise, find the time(s) that the projectile will (a) reach a height of 80 ft and (b) return to the ground for the given value of v_0. Round answers to the nearest hun-dredth if necessary. v_0=96

Solve each equation using the square root property. See Example 2. x^2 = 121

Solve each equation in Exercises 47–64 by completing the square.

x^2 - 5x + 6 = 0

Solve each equation in Exercises 1 - 14 by factoring.

x^2 - 3x - 10 = 0

Algebra II: Quadratic Equations - Factoring (Level 8 of 10)

Solving Quadratic Equations using the Quadratic Formula - Example 2, Complex Solutions

Example 4: Applying the quadratic formula | Quadratic equations

Solve each equation in Exercises 1 - 14 by factoring.

3x^2 + 12x = 0

Solve each equation in Exercises 83–108 by the method of your choice.

9 - 6x + x^2 = 0

Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer, then x+1 represents the next consecutive integer.If x represents an even integer, then x+2 represents the next consecutive even integer.If x represents an odd integer, then x+2 represents the next consecutive odd integer. The sum of the squares of two consecutive even integers is 52. Find the integers.

For each equation, (a) solve for x in terms of y.. See Example 8. 4x^2 - 2xy + 3y^2 = 2

Answer each question. Find the values of a, b, and c for which the quadratic equation. 
ax^2 + bx + c = 0 has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.) i, -i

Answer each question. Answer each question. Answer each question. Unknown NumbersUse the following facts.If x represents an integer, then x+1 represents the next consecutive integer.If x represents an even integer, then x+2 represents the next consecutive even integer.If x represents an odd integer, then x+2 represents the next consecutive odd integer. Find two consecutive even integers whose product is 168.

Solving for 'X' ;  Quadratic Equations Involving the Pythagorean Theorem - Example 2

Solve each equation in Exercises 83–108 by the method of your choice.

(3x - 4)^2 = 16

Solving Trigonometric Equations Using the Quadratic Formula - Example 3

Solve each equation in Exercises 83–108 by the method of your choice.

4x^2 - 16 = 0

Solve each equation in Exercises 83–108 by the method of your choice.

2x/(x - 3) + 6/(x + 3) = - 28/(x^2 - 9)

Solve each equation in Exercises 47–64 by completing the square.

4x^2 - 4x - 1 = 0

Match the equation in Column I with its solution(s) in Column II. x^2 = 25

Solve each equation in Exercises 47–64 by completing the square.

x^2 + 6x = 7

In Exercises 109–114, find the x-intercept(s) of the graph of each equation. Use the x-intercepts to match the equation with its graph. The graphs are shown in [- 10, 10, 1] by [- 10, 10, 1] viewing rectangles and labeled (a) through (f). 

y = x^2 - 4x - 5

Solving Quadratic Equations by Completing the Square

Solve each equation in Exercises 73–81 by the method of your choice. 3x^2-7x+1 =0

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9. 9x^2 + 11x + 4 = 0

In Exercises 115–122, find all values of x satisfying the given conditions.

y1 = - x^2 + 4x - 2, y2 = - 3x^2 + x - 1, and y1 - y2 = 0

In Exercises 115–122, find all values of x satisfying the given conditions.

y = 5x^2 + 3x and y = 2

In Exercises 127–130, solve each equation by the method of your choice.

1/(x^2 - 3x + 2) = 1/(x + 2) + 5/(x^2 - 4)

Solve each cubic equation using factoring and the quadratic formula. See Example 7. 
x^3 + 64 = 0

Write a quadratic equation in general form whose solution set is {- 3, 5}.

Solve quadratic equations by completing the square

Solve Quadratic Equations by Factorization / Example 2.6

Solving Word Problems involving Distance, Rate, and Time Using Quadratics - Example 3

Solve each equation using completing the square. See Examples 3 and 4. 3x^2 - 9x + 7 = 0

Solve each equation in Exercises 83–108 by the method of your choice.

(2x + 3)(x + 4) = 1

For each equation, (a) solve for x in terms of y. See Example 8. 2x^2 + 4xy - 3y^2 = 2

 Solving Quadratic Equations Using the Square Root Method / Example 2.1

 Summary of Methods

Solve a quadratic equation: completing the square (2)

Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) See Example 9. 8x^2 - 72 = 0

Solve each equation for the specified variable. (Assume no denominators are 0.) See Example 8. h = -16t^2+v_0t+s_0, for t

Solve a quadratic equation: taking a square root

 Solve Quadratic Equations by Factoring a Trinomial / Example 2.9

In Exercises 35–46, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.

x^2 + 3x

Completing the Square to Solve Quadratic Equations: More Examples - 2

Completing the Square to Solve Quadratic Equations: More Examples - 5

In Exercises 115–122, find all values of x satisfying the given conditions.

y1 = 2x^2 + 5x - 4, y2 = - x^2 + 15x - 10, and y1 - y2 = 0

Solve each equation using the square root property. See Example 2. 27 - x^2 = 0

Solve each equation in Exercises 15–34 by the square root property.

(x + 2)^2 = 25

Exercises 73–75 will help you prepare for the material covered in the next section.

Multiply: (7 - 3x)(- 2 - 5x)

In Exercises 115–122, find all values of x satisfying the given conditions.

y1 = 2x/(x + 2), y2 = 3/(x + 4), and y1 + y2 = 1

Use Choices A–D to answer each question. A. 3x^2 - 17x - 6 = 0 B. (2x + 5)^2 = 7 C. x^2 + x = 12 D. (3x - 1)(x - 7) = 0 
Which equation is set up for direct use of the zero-factor property? Solve it

A Projectile Problem Using Quadratics - Example 1

Solving Quadratics by Completing the Square

Solve each equation using the zero-factor property. See Example 1. 36x^2 + 60x + 25 = 0

In Exercises 35–46, determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial.

x^2 + 12x

Solve each equation in Exercises 47–64 by completing the square.

x^2 + 3x - 1 = 0

Answer each question. Find the values of a, b, and c for which the quadratic equation. 
ax^2 + bx + c = 0 has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.) -3, 2

Determine the best method to solve a quadratic equation

In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation.

x^2 - 3x - 7 = 0

Solve each equation using completing the square. See Examples 3 and 4. 2x^2 + x = 10

 Solving Quadratic Equations by Completing the Square / Example 2.3

Word Problems Using the Pythagorean Theorem - Example 1

In Exercises 127–130, solve each equation by the method of your choice.

√2 x^2 + 3x - 2√2 = 0

Factoring Trinomials (A quadratic Trinomial) by Trial and Error

❖ Completing the Square - Solving Quadratic Equations ❖

Solve each equation in Exercises 65–74 using the quadratic formula.

3x^2 - 3x - 4 = 0

Deriving the Quadratic Formula

Solve each equation in Exercises 66–67 by completing the square. 3x^2 -12x+11= 0

Solve by completing the square: 2x² – 5x + 1 = 0.

Solve each equation using the quadratic formula. See Examples 5 and 6. (4x - 1)(x + 2) = 4x

Match the equation in Column I with its solution(s) in Column II. x^2 + 5 = 0

Solve each equation using the quadratic formula. See Examples 5 and 6. x^2 - x - 1 = 0

Understand the discriminant and nature of the roots

Completing the Square to Solve Quadratic Equations: More Examples - 6

Solve each equation in Exercises 83–108 by the method of your choice.

1/x + 1/(x + 2) = 1/3