Factorizing Quadratic Equation
Question 1. Factorise x² +9x+14
Solution
To factorize x² + 9x + 14, we need to find two numbers whose product is 14 and whose sum is 9.
We can easily see that 2 and 7 are the two numbers we’re looking for, since 2 × 7 = 14 and 2 + 7 = 9.
So we can rewrite x² + 9x + 14 as:
x² + 2x + 7x + 14
Now we group the terms:
(x² + 2x) + (7x + 14)
We can factor out x from the first group and 7 from the second group:
x(x + 2) + 7(x + 2)
Now we have a common factor of (x + 2):
(x + 2)(x + 7)
So the factorization of x² + 9x + 14 is:
(x + 2)(x + 7)
Question 2. Factorize 4x³y – 2xy²
Solution
To factorize 4x³y – 2xy², we need to find the greatest common factor (GCF) of the two terms, and then factor it out. The GCF of 4x³y and 2xy² is 2xy, so we can write:
4x³y – 2xy² = 2xy(2x² – y)
So the factorization of 4x³y – 2xy² is:
2xy(2x² – y)
Question 3. A man’s salary is # 3,200 per month. If he receives a 4% increase, find his new salary.
Solution
To find the man’s new salary after a 4% increase, we can use the following formula:
New Salary = Old Salary + (Percent Increase × Old Salary)
where Percent Increase is the increase expressed as a decimal (i.e., 4% = 0.04).
Using this formula, we can substitute the given values to get:
New Salary = 3200 + (0.04 × 3200)
New Salary = 3200 + 128
New Salary = 3328
Therefore, the man’s new salary is # 3,328 per month.
Question 4. Make a bearing sketch of the following angles: (a) 080° (b) 124° (c) 315′ (d) 222°
Solution
To make a bearing sketch of an angle, we draw a line representing the direction the angle is pointing, and then indicate the angle with respect to a reference direction (usually North).
(a) 080°: We draw a line pointing towards 80° clockwise from North, and label it “080°”.
N
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W—-O—-> 080°
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S
(b) 124°: We draw a line pointing towards 124° clockwise from North, and label it “124°
N
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W—-O—-> 124°
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S
(c) 315°: We draw a line pointing towards 315° clockwise from North, and label it “315°”.
N
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W—-O—-> 315°
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S
(d) 222°: We draw a line pointing towards 222° clockwise from North, and label it “222°”.
N
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W—-O—-> 222°
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S
Note that in (c), the angle is expressed in degrees and minutes, so we convert 315′ to decimal degrees by dividing by 60: 315/60 = 5.25 degrees. Therefore, the angle points towards 5.25° clockwise from the West direction.
Question 4. Simplify 3k/5 + k/2
To simplify 3k/5 + k/2, we need to find a common denominator for the two fractions. The smallest common multiple of 5 and 2 is 10, so we can rewrite the fractions with 10 as the denominator:
3k/5 = 6k/10
k/2 = 5k/10
Now we can add the two fractions:
6k/10 + 5k/10 = (6k + 5k) / 10 = 11k/10
So the simplified form of 3k/5 + k/2 is:
11k/10
Steps in solving Quadratic equation
Here are some key steps to factorize quadratic equations that I would explain to JSS 2 students:
- Write the quadratic equation in the form ax² + bx + c = 0, where a, b, and c are constants.
- Identify the values of a, b, and c in the equation.
- Find the product of a and c, and then find two factors of this product that add up to b. If you cannot find such factors, then the equation cannot be factored using real numbers.
- Write the factors of the product found in step 3 in the form (mx + p) and (nx + q), where m, n, p, and q are constants.
- Rewrite the quadratic equation in the form (mx + p)(nx + q) = 0, using the factors found in step 4.
- Use the distributive property to expand the equation in step 5, and simplify by combining like terms.
- Set each factor equal to zero, and solve for x. This will give you the roots or solutions of the quadratic equation.
- Check your work by multiplying the factors you found in step 4, and verifying that the result is equal to the original quadratic equation in step 1.
With practice, you will become more familiar with these steps and be able to factorize quadratic equations quickly and accurately
Worked Samples
Example 1: Factorize x² + 4x + 4 Solution:
- Write the quadratic equation in the form ax² + bx + c = 0, where a, b, and c are constants. Here, a = 1, b = 4, and c = 4, so the equation becomes x² + 4x + 4 = 0.
- Find the product of a and c: a x c = 1 x 4 = 4.
- Find two factors of 4 that add up to b = 4. These factors are 2 and 2.
- Write the factors in the form (mx + p) and (nx + q): (x + 2)(x + 2).
- Rewrite the quadratic equation using the factors: (x + 2)(x + 2) = 0.
- Simplify by expanding the equation: x² + 4x + 4 = 0.
- Set each factor equal to zero, and solve for x: x + 2 = 0 or x + 2 = 0. The roots are x = -2, -2.
- Check your work by multiplying the factors: (x + 2)(x + 2) = x² + 4x + 4, which is the same as the original equation. So, the factorization is (x + 2)(x + 2).
Example 2: Factorize 2x² + 7x + 3 Solution:
- Write the quadratic equation in the form ax² + bx + c = 0, where a, b, and c are constants. Here, a = 2, b = 7, and c = 3, so the equation becomes 2x² + 7x + 3 = 0.
- Find the product of a and c: a x c = 2 x 3 = 6.
- Find two factors of 6 that add up to b = 7. These factors are 1 and 6.
- Write the factors in the form (mx + p) and (nx + q): (2x + 1)(x + 3).
- Rewrite the quadratic equation using the factors: (2x + 1)(x + 3) = 0.
- Simplify by expanding the equation: 2x² + 7x + 3 = 0.
- Set each factor equal to zero, and solve for x: 2x + 1 = 0 or x + 3 = 0. The roots are x = -1/2, -3.
- Check your work by multiplying the factors: (2x + 1)(x + 3) = 2x² + 7x + 3, which is the same as the original equation. So, the factorization is (2x + 1)(x + 3).
Example 3: Factorize 3x² – 10x – 8 Solution:
- Write the quadratic equation in the form ax² + bx + c = 0, where a, b, and c are constants. Here, a = 3, b = -10, and c = -8, so the equation becomes 3x² – 10x – 8 = 0.
- Find the product of a and c: a x c = 3 x -8 = -24.
- Find two factors of -24 that add up to b = -10. These factors are -12 and 2.
- Write the factors in the form (mx + p) and (nx + q): (3x – 4)(x – 2).
- Rewrite the quadratic equation using the factors: (3x – 4)(x – 2) = 0.
- Simplify by expanding the equation: 3x² – 10x – 8 = 0.
- Set each factor equal to zero, and solve for x: 3x – 4 = 0 or x – 2 = 0. The roots are x = 4/3, 2.
- Check your work by multiplying the factors: (3x – 4)(x – 2) = 3x² – 10x – 8, which is the same as the original equation. So, the factorization is (3x – 4)(x – 2).
Example 4: Factorize 2x² – 5x – 3 Solution:
- Write the quadratic equation in the form ax² + bx + c = 0, where a, b, and c are constants. Here, a = 2, b = -5, and c = -3, so the equation becomes 2x² – 5x – 3 = 0.
- Find the product of a and c: a x c = 2 x -3 = -6.
- Find two factors of -6 that add up to b = -5. These factors are -2 and 3.
- Write the factors in the form (mx + p) and (nx + q): (2x – 3)(x + 1).
- Rewrite the quadratic equation using the factors: (2x – 3)(x + 1) = 0.
- Simplify by expanding the equation: 2x² – 5x – 3 = 0.
- Set each factor equal to zero, and solve for x: 2x – 3 = 0 or x + 1 = 0. The roots are x = 3/2, -1.
- Check your work by multiplying the factors: (2x – 3)(x + 1) = 2x² – 5x – 3, which is the same as the original equation. So, the factorization is (2x – 3)(x + 1).
Example 5: Factorize x² – 6x + 5 Solution:
- Write the quadratic equation in the form ax² + bx + c = 0, where a, b, and c are constants. Here, a = 1, b = -6, and c = 5, so the equation becomes x² – 6x + 5 = 0.
- Find the product of a and c: a x c = 1 x 5 = 5.
- Find two factors of 5 that add up to b = -6. These factors are -1 and -5.
- Write the factors in the form (mx + p) and (nx + q): (x – 1)(x – 5).
- Rewrite the quadratic equation using the factors: (x – 1)(x – 5) = 0.
- Simplify by expanding the equation: x² – 6x + 5 = 0
Evaluation :
1. To factorize a quadratic equation, write it in the form ___ + ___ + ___ = 0.
a. ax + b + c
b. ax² + bx + c
c. ax² – bx – c
2. The smallest common multiple of 2 and 3 is ___.
a. 2
b. 3
c. 6
3. If a quadratic equation cannot be factored using real numbers, it is said to have ___ roots.
a. real
b. complex
c. imaginary
4. To find the product of a and c in a quadratic equation, you multiply ___ and ___.
a. a and b
b. a and c
c. b and c
5. To factorize x² + 5x + 6, you need to find two factors of ___ that add up to ___.
a. 6, 5
b. 6, 6
c. 5, 5
6. The factorization of 3x² – 12x + 9 is ___.
a. 3(x – 3)(x + 3)
b. 3(x – 3)(x – 3)
c. (3x – 3)(x – 3)
7. To factorize 4x² – 4x – 3, you need to find two factors of ___ that add up to ___.
a. -12, 4
b. -12, -4
c. -3, -4
8. The factorization of x² – 7x + 12 is ___.
a. (x – 3)(x + 4)
b. (x – 4)(x – 3)
c. (x + 3)(x – 4)
9. To factorize 2x² – 5x – 3, you need to find two factors of ___ that add up to ___.
a. -6, -5
b. -6, 5
c. 6, -5
10. The factorization of x² + 6x + 8 is ___.
a. (x + 2)(x + 4)
b. (x – 2)(x + 4)
c. (x – 2)(x – 4)
Answers:
1. b, 2. c, 3. b, 4. b, 5. a, 6. b, 7. b, 8. b, 9. a, 10. a
[mediator_tech]
Lesson Plan Presentation
Title: Factorizing Quadratic Equations
Objectives:
– Define quadratic equations
– Explain the process of factorizing quadratic equations
– Solve quadratic equations by factorization
Materials Needed:
– Whiteboard and markers
– Worksheet with practice problems
– Calculator (optional)
Introduction (5 minutes):
– Review what a quadratic equation is and provide examples
– Explain that in this lesson, we will be learning how to factorize quadratic equations.
Body (35 minutes):
Step 1: Introduce the process of factorizing quadratic equations
– Write a quadratic equation on the board and show how to factorize it
– Explain that the process involves finding two factors of the constant term that add up to the coefficient of the middle term
– Write out the steps for factorizing a quadratic equation (as shown earlier in this conversation)
Step 2: Demonstrate factorization with examples
– Provide examples of quadratic equations and guide students through the process of factorizing them
– Emphasize the importance of checking the work by multiplying the factors to ensure they equal the original equation
– Answer questions and clarify any misunderstandings
Step 3: Provide practice problems
– Distribute a worksheet with practice problems on factorizing quadratic equations
– Allow students to work on the problems individually or in pairs
– Circulate the room to provide assistance and feedback
Step 4: Review answers and check work
– Review the practice problems as a class and have students share their answers
– Check the work of the students to ensure they have factored the equations correctly
– Provide feedback and answer any questions
Conclusion (5 minutes):
– Recap the process of factorizing quadratic equations
– Emphasize the importance of checking the work and practicing regularly
– Encourage students to ask questions and seek help when needed.
Assessment:
– Evaluate students’ comprehension through class participation, worksheet completion, and answering questions during the lesson
– Provide feedback and offer additional resources or support to students who need it.
Extension activities:
– Challenge students to solve more complex quadratic equations by factorization
– Assign homework that involves factoring quadratic equations
– Encourage students to research real-world applications of quadratic equations and share their findings with the class.
Note: The time for each section can be adjusted based on the pace and needs of the students.
Evaluation :
1. What is a quadratic equation?
2. What is the general form of a quadratic equation?
3. What is the process of factorizing a quadratic equation?
4. How do you find the factors of a quadratic equation?
5. What is the difference between factoring and solving a quadratic equation?
6. Can all quadratic equations be factored using real numbers? Why or why not?
7. What are the roots of a quadratic equation?
8. How can you check your work when factorizing a quadratic equation?
9. What are some real-world applications of quadratic equations?
10. How can you practice and improve your skills in factorizing quadratic equations?
[mediator_tech]
Answers:
1. A quadratic equation is an equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.
2. The general form of a quadratic equation is ax^2 + bx + c = 0.
3. The process of factorizing a quadratic equation involves finding two factors of the constant term that add up to the coefficient of the middle term, and writing the equation as the product of these factors.
4. You find the factors of a quadratic equation by finding two numbers that multiply to the constant term and add up to the coefficient of the middle term.
5. Factoring a quadratic equation involves writing it in the form (mx + p)(nx + q) = 0, while solving a quadratic equation involves finding the roots or solutions of the equation.
6. Not all quadratic equations can be factored using real numbers. Some equations have roots that are complex or imaginary numbers.
7. The roots of a quadratic equation are the values of x that satisfy the equation and make it equal to zero.
8. You can check your work when factorizing a quadratic equation by multiplying the factors and verifying that the result is equal to the original equation.
9. Quadratic equations have many real-world applications, such as in physics, engineering, and finance.
10. You can practice and improve your skills in factorizing quadratic equations by doing practice problems, working with a tutor or study group, and seeking feedback and help when needed.
[mediator_tech]
Weekly Assessment /Test
1. To factorize a quadratic equation, write it in the form ___ + ___ + ___ = 0.
a. ax + b + c
b. ax² + bx + c
c. ax² – bx – c
2. The process of factorizing a quadratic equation involves finding two factors of the ___ term that add up to the coefficient of the ___ term.
a. constant, middle
b. middle, constant
c. variable, constant
3. The product of the factors in a quadratic equation is equal to ___.
a. the sum of the coefficients
b. the product of the coefficients
c. the constant term
4. To factorize x² + 4x + 4, you need to find two factors of ___ that add up to ___.
a. 2, 2
b. 4, 4
c. 4, 2
5. The factorization of 3x² – 12x + 9 is ___.
a. 3(x – 3)(x + 3)
b. 3(x – 3)(x – 3)
c. (3x – 3)(x – 3)
6. To factorize 4x² – 4x – 3, you need to find two factors of ___ that add up to ___.
a. -12, 4
b. -12, -4
c. -3, -4
7. The factorization of x² – 7x + 12 is ___.
a. (x – 3)(x + 4)
b. (x – 4)(x – 3)
c. (x + 3)(x – 4)
8. If a quadratic equation cannot be factored using real numbers, it has ___ roots.
a. real
b. complex
c. imaginary
9. The factorization of 2x² + 5x – 3 is ___.
a. (2x – 1)(x + 3)
b. (2x + 3)(x – 1)
c. (2x + 1)(x – 3)
10. When factorizing a quadratic equation, you should always check your work by ___.
a. solving the equation
b. adding the coefficients
c. multiplying the factors
Answers:
1. b, 2. a, 3. c, 4. a, 5. b, 6. c, 7. a, 8. b, 9. b, 10. c
