Quadratic Equations: A Comprehensive Module for Grade 9

• $3x^2 + 2x = 9$

  • Standard Form: $3x^2 + 2x – 9 = 0$ with $a = 3$, $b = 2$, and $c = −9$

• b. $(x − 5) (2x + 3) = 7$

  • Simplify: $2x^2 – 7x – 15 = 7 → 2x^2 – 7x – 8 = 0$ with $a = 2$, $b = −7$, and $c = −8$

• c. $2x^2 – 15 = 2(x^2 + 7x)$

  • Simplify: $2x^2 – 15 = 2x^2 + 14x → −14x – 15 = 0$. This is a linear equation, not quadratic.

Check Your Understanding

• Complete the table (not provided in text but implied) to practice identifying $a$, $b$, and $c$ in quadratic equations.

Examples for the Table:

• $x ( 2x – 7) = 0 → 2x^2 – 7x = 0$ (a = 2, b = -7, c = 0)

• $(x + 5)( x + 9) = 20$

• $(2x – 1)( x + 3) = -2x$

• $(x + 6)^2 = 12x$

• $(5x + 4) (5x – 4) = 0$

• $(3x – 8)(x + 2) = 2x^2$

Example 2: Representing Situations with Quadratic Equations

• Represent each situation using a quadratic equation.

Examples:

• a. A rectangular prism has a height of 3 m. Its length is three times its width. The volume of the prism is 18 $m^3$.

  • Let $w$ = width, $3w$ = length

  • Volume: $V = lwh → 18 = (3w)(w)(3) → 18 = 9w^2 → 9w^2 – 18 = 0$

• b. Jody wants each window to have an area of 315 $cm^2$, and the length to be 6 cm more than the width.

  • Let $x$ = width, $x + 6$ = length

  • Area: $A = lw → 315 = x(x+6) → x^2 + 6x – 315 = 0$

• c. The product of two numbers is 48 and their sum is 16.

  • Let $x$ be the 1st number, $16 − x$ be the 2nd number

  • $x(16 − x) = 48 → 16x − x^2 = 48 → −x^2 + 16x − 48 = 0$

• d. The product of two consecutive even integers is 288.

  • Let $x$ be the 1st even number, $x + 2$ be the 2nd even number

  • $x(x + 2) = 288 → x^2 + 2x = 288 → x^2 + 2x − 288 = 0$

What's More: Test Yourself!

Part A: Identifying Quadratic Equations

Which of these equations describe a quadratic equation?

1. $A = πr^2$

2. $x + 4x^2 = 0$

3. $(x − 2)^2 − 5 = 0$

4. $(x + 3) + 8 = 0$

5. $x^2 = 0$

Part B: Representing Situations

Represent the following situations using quadratic equation in standard form.

1. If the square of a number is added to 8 times the number, the result is 100.

2. Mr. Apoloan wants to lay out a rectangular playground with an area of 30 square feet. The desired length will be 7 times the width.

What I Can Do: Critical Thinking

Which of the following is a quadratic equation? For those that are, identify the value of a, b and c.

1. $x^2 – 6x + 2 = 0$

2. $3x + 5 = 0$

3. $2x^2 + 7x = 15$

4. $(x + 1)^2 = 2(x – 3)$

5. $x^2 + 2x + 1 = 5x + 6$

6. $(x − 1) (x + 2) = x(x + 5)$

7. $(x + 2) (x – 3) = 5$

8. $x(x^2 + 3x – 10) = 0$

9. $(x – 1)^2 + 3 = 2x + 1$

10. $(x + 2)^3 = x(x^2 – 10x + 25)$

What I Have Learned

A quadratic equation in one variable is a second-degree equation that can be written in the form $ax^2 + bx + c = 0$, where a, b , and c are real numbers and $a ≠ 0$.

Assessment

Write the letter of the correct answer on your answer sheet.

1. Given a, b, and c are real numbers and $a ≠ 0$, which of the following is the standard form of quadratic equation?
   A. $ax^2 + bx + c = 0$
   B. $ax + b + c = 0$
   C. $a = bx + c$
   D. $ax + b^2 + c = 0$

2. Which of the following is a quadratic equation?
   A. $3x + 5 = 0$
   B. $(x + 1) (x – 2) = x^2$
   C. $(x + 2) (x – 3) = 5$
   D. $2x^3 + x^2 – 5 = 0$

3. Given $(x – 1)^2 + 3 = 2x + 1$, find the value of the constant term.
   A. 1
   B. 3
   C. -1
   D. -3

4. Which of the following is equivalent to $x^2 + 4x – 7 = 0$?
   A. $(x + 1) (x – 7) = 0$
   B. $(x + 4) (x – 3) = 5$
   C. $(x + 5)(x – 3) = 6$
   D. $(x + 1)(x + 3) = 10$

5. When the quadratic equation $(x + 5) (2x – 3) = 2(x + 1)$ is written in standard form, what are the values of a, b, and c.
   A. $a = 4, b = 19, c = −17$
   B. $a = 2, b = 5, c = −17$
   C. $a = 1, b = −5, c = 17$
   D. $a = 2, b = 10, c = 17$

6. Which of the following situations illustrates quadratic equation?
   A. A garden’s length is 7m longer that its width and the area is 18 square meter.
   B. A lot cost P4,000 per square meter and the area is 120 square meters.
   C. Joey paid at least P2,000 for the shirt and pants. The cost of pants is P700 more than the shirt.
   D. The length of a rectangular board is 3m longer than the width and the perimeter is 25m.

7. Express the statement in quadratic equation: “A rectangular field with an area of 120$m^2$ has its length 12m. longer than the width.”
   A. $x^2 − 12x − 120 = 0$
   B. $x^2 + 12x + 120 = 0$
   C. $x^2 + 12x – 120 = 0$
   D. $x^2 – 12x + 120 = 0$

8. Ryan is constructing a play house. He wants each window to have an area of 315 $cm^2$. Which of the statements below forms a quadratic equation?
   A. The length of each window is 4 less than the square of the width.
   B. The length of each window is 6 cm more than the width.
   C. The area of the window is twice its width.
   D. The sum of the length and width is twice its area.

9. The area of a square with side s is 144 $cm^2$. Which quadratic equation represents the area of the square?
   A. $4s = 144$
   B. $s^2 = 144$
   C. $s (s + 1) = 144$
   D. $(s + 1)^2 = 144$

10. A rectangular lot has an area of 132$m^2$ and a perimeter of 46m. Which of the following quadratic equations illustrates the given situation?
    A. $x^2 − 23x + 132 = 0$
    B. $x^2 + 23x + 132 = 0$
    C. $x^2 + 46x – 132 = 0$
    D. $x^2 – 46x + 132 = 0$

What I Can Do

After dealing with examples of quadratic equations and possible situation that can represent quadratic equation, think and reflect. How will quadratic equations help you in solving real-life problems and making decisions?

Write a real-life word problem that will lead you to forming quadratic equation. Provide an illustration of the problem and guide questions such that it will be expressed as quadratic equation in standard form.

Additional Activities

You may also check the following link for your reference and further learnings on illustrating quadratic equations.

• https://examples.yourdictionary.com/examples-of-quadratic-equation.html
• https://www.mathsisfun.com/algebra/quadratic-equation.html
• https://mathbitsnotebook.com/Algebra1/Quadratics/QDquadequations.html
• https://www.math-only-math.com/introduction-to-quadratic-equation.html https://www.mathsisfun.com/algebra/quadratic-equation.html
• https://bit.ly/2W94SG7

Problem – Based Worksheet

The Vegetable Garden

Mang Toti joined an organization that encourages people to go into backyard farming. He owns a rectangular piece of vacant lot adjacent to his house. He is planning to convert his lot into vegetable garden for additional income.

Let’s Analyze

1. If the area of the vacant lot is 12$m^2$ and the length is 4m longer than the width, how will you represent the length if the width is represented by w?

2. What expression represents the area of the of the vacant lot? _

3. What quadratic equation in general form represents the situation? _

4. Supposing Mang Toti bought 16m of fencing materials and plan to place it all around his rectangular land, what equation describes the situation? __

Answer Key

The answer key is provided for various sections, including 'What’s In', 'What I Can Do', 'What’s More', 'What I Know' and the Assessment.

References

• Dilao, S. J., & Bernabe, J. G. (2009). Intermediate Algebra.pp 52-54. Quezon CIty: SD Publicaton.

• MacKeague, Charles, Intermediate Algebra, Concepts and Graphs. Saunders College Publishing, USA

• Mathematics 9 Learner’s Material, Department of Education

• Orines, Fernando B., et. al. Next Century Mathematics (Intermediate Algebra. Quezon City, Philippines: Phoenix Publishing House

• Oronce, Orlando A. & Mendoza, Marilyn O. Exploring Mathematics II (Intermediate Algebra). Manila, Philippines: Rex Book Store Inc.

• Ogena, Ester, et. al. Our Math Grade 9. Mc Graw Hill, Vibal Group. Inc.

• https://www.freepik.com/free-vector/woman-with-long-hair-teaching-online_7707557.htm

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