TEKS A.8A Worksheets — Algebra I Solve quadratic equations having real solutions by

Adrian is planning a fishing trip to Corpus Christi, Texas. He wants to rent a boat that can hold a maximum of 1200 pounds. If the boat rental company charges a flat fee of $150 plus $30 per hour of rental, which equation can be used to find the number of hours, h, he can rent the boat if he wants to stay within his budget of $450?

1. 30h + 150 = 450 ✓
2. 30h - 150 = 450
3. 150h + 30 = 450
4. 150h - 30 = 450

Why: To find the number of hours Adrian can rent the boat while staying within his budget of $450, we start with the total cost equation, which includes the flat fee and the hourly rate: Cost = Flat Fee + Hourly Rate * Hours, or C = 150 + 30h. Setting this equal to his budget gives us the equation: 30h + 150 = 450. From here, Adrian can solve for h to determine the maximum number of hours he can afford to rent the boat.

Victoria is designing a small garden in her backyard located in Lufkin, Texas. She wants the garden to have a width that is 2 feet less than its length. If the area of the garden is represented by the equation x(x - 2) = 48, where x is the length in feet, what is the length of the garden?

1. 8 ✓
2. 6
3. 10
4. 4

Why: To find the length of the garden, we start with the equation x(x - 2) = 48. Expanding this, we get x^2 - 2x = 48. Rearranging gives us x^2 - 2x - 48 = 0. Next, we can factor the quadratic: (x - 8)(x + 6) = 0. Setting each factor to zero gives us x - 8 = 0 or x + 6 = 0. Thus, x = 8 or x = -6. Since a length cannot be negative, the length of the garden is 8 feet.

Adrian is planting two types of flowers in his backyard in Lubbock, Texas. He has x number of red flowers and y number of blue flowers. The total number of flowers he wants to plant is 50. Additionally, the number of red flowers should be twice the number of blue flowers. Which of the following equations can be used to find the number of red flowers (x) and blue flowers (y)?

1. x + y = 50 and x = 2y ✓
2. x + y = 50 and y = 2x
3. x + 2y = 50 and y = x
4. 2x + y = 50 and x = y

Why: To solve for the number of red flowers (x) and blue flowers (y), we can establish two equations based on the information given. The first equation states that the total number of flowers is 50, which can be written as x + y = 50. The second condition states that the number of red flowers is twice the number of blue flowers, which can be expressed as x = 2y. Therefore, the correct equations are x + y = 50 and x = 2y.

During the San Antonio Stock Show, Mateo is selling handmade leather goods for his family business. His profit P, in dollars, can be modeled by the quadratic equation P(x) = -2(x - 5)(x - 15), where x is the number of items sold. How many items does Mateo need to sell to maximize his profit?

1. 10 ✓
2. 5
3. 15
4. 20

Why: To maximize profit, we need to determine the vertex of the parabola represented by the equation P(x) = -2(x - 5)(x - 15). The x-coordinate of the vertex, which gives the maximum point, can be found using the formula x = (p + q) / 2, where p and q are the roots of the equation. Here, p = 5 and q = 15, so x = (5 + 15) / 2 = 10. Therefore, Mateo needs to sell 10 items to achieve maximum profit.

Fatima is helping her family plant citrus trees in their McAllen backyard. They plan to plant a total of 72 trees this season, and they are using two types of trees: orange trees and lemon trees. The number of orange trees, represented by x, is twice the number of lemon trees. If they plant a total of 72 trees, how many orange trees will they plant?

1. 48 ✓
2. 24
3. 36
4. 12

Why: Let the number of lemon trees be represented as y. Since the number of orange trees is twice the number of lemon trees, we can say x = 2y. The total number of trees is given by the equation x + y = 72. Substituting the value of x gives us 2y + y = 72. This simplifies to 3y = 72, so y = 72 / 3 = 24 (lemon trees). Using the relationship x = 2y, we find x = 2 * 24 = 48 (orange trees). Therefore, the number of orange trees Fatima will plant is 48.

Jayden is organizing a fundraiser in New Braunfels to plant mesquite trees in the local park. The cost to plant x trees can be modeled by the equation x^2 - 5x - 14 = 0. How many trees can Jayden afford to plant if he can only spend money on whole trees?

1. 7 ✓
2. 2
3. 3
4. 5

Why: To solve the quadratic equation x^2 - 5x - 14 = 0, we need to factor it. Looking for two numbers that multiply to -14 and add to -5, we find -7 and 2. This gives us the factors (x - 7)(x + 2) = 0. Setting each factor equal to zero, we have x - 7 = 0 or x + 2 = 0, which gives us x = 7 or x = -2. Since Jayden can only afford whole trees, the only valid solution is x = 7.

Emma wants to plant a garden in her backyard in Lubbock, Texas. The area of the rectangular garden is represented by the equation x^2 - 5x - 14 = 0, where x is the length of one side of the garden. What is the length of the side of the garden that Emma can use, rounded to the nearest whole number?

1. 7 ✓
2. 2
3. 5
4. 14

Why: To solve for x in the equation x^2 - 5x - 14 = 0, we can factor it into (x - 7)(x + 2) = 0. Setting each factor equal to zero gives us x - 7 = 0, which simplifies to x = 7, and x + 2 = 0, which gives x = -2. Since length cannot be negative, the only valid solution is x = 7. Therefore, the length of the side of the garden that Emma can use is 7.

Rohit is planning a school trip to Enchanted Rock and needs to raise money for transportation. He decides to sell pecan pies for $10 each. After selling x pies, he has a total of $50. If he spends $30 on supplies, which of the following equations represents the situation where Rohit has a profit of $20 after selling the pies?

1. 10x - 30 = 20 ✓
2. 10x - 50 = 20
3. 10x + 30 = 20
4. 10x - 30 = 50

Why: To find the profit, we first consider Rohit's revenue from selling pies, which is 10x dollars for x pies. He has spent $30 on supplies, so his total expenses are $30. To have a profit of $20, we set up the equation: revenue - expenses = profit. This means 10x - 30 = 20. Therefore, the correct equation is 10x - 30 = 20.