This document provides instructions on solving quadratic equations using different methods: factoring, extracting square roots, the quadratic formula, and determining the discriminant. It explains that if the product of two terms is zero, one of the terms must be zero. This is used to solve equations by setting each factor equal to zero. Examples are provided to demonstrate solving quadratic equations by factoring, taking the square root of both sides, and using the discriminant to determine the number of solutions.
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
Quadratic Equations
In One Variable
1. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers
2.Types of Quadratic Equations
Complete Quadratic
3x2 + 5x + 6 = 0
Incomplete/Pure Quadratic Equation
3x2 - 6 = 0
3.Solving an Incomplete Quadratic
4.Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2
5.Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3
6.Solving Quadratic Equation
7.By Factoring
Place all terms in the left member of the equation, so that the right member is zero.
Factor the left member.
Set each factor that contains the unknown equal to zero.
Solve each of the simple equations thus formed.
Check the answers by substituting them in the original equation.
8.Example: x² = 6x - 8
Solution:
x² = 6x – 8
x² - 6x + 8 = 0
(x – 4)(x – 2) = 0
x – 4 = 0 | x – 2 = 0
x = 4 x = 2
9.By Completing the Square
Write the equation with the variable terms in the left member and the constant term in the right member.
If the coefficient of x² is not 1, divide every term by this coefficient so as to make the coefficient of x² equal to 1.
Take one-half the coefficient of x, square this quantity, and add the result to both members.
Find the square root of both members, placing a ± sign before the square root of the right member.
Solve the resulting equation for x.
10.Example: x² - 8x + 7 = 0
11.By Quadratic Formula
Example: 3x² - 2x - 7 = 0
12.Solve the following:
1. x² - 15x – 56 = 0
2. 7x² = 2x + 6
3. 9x² - 3x + 8 = 0
4. 8x² + 9x -144 = 0
5. 2x² - 3 + 12x
13.Activity:
Solve the following quadratic formula.
By Factoring By Quadratic Formula
1. x² - 5x + 6 = 0 1. x² - 7x + 6 = 0
2. 3 x² = x + 2 2. 10 x² - 13x – 3 = 0
3. 2 x² - 11x + 12 = 0 3. x (5x – 4) = 2
By Completing the Square
1. x² + 6x + 5 = 0
2. x² - 8x + 3 = 0
3. 2 x² + 3x – 5 = 0
Mathematics 9 Lesson 1-B: Solving Quadratic Equations using Quadratic FormulaJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using the Quadratic Formula. It also discusses the steps in solving quadratic equations using the method of Quadratic Formula.
Hello everyone! I hope this short powerpoint presentation can help you in understanding quadratic formula, especially for those students that are having hard time to cope up in this topic.
IT'S A PRESENTATION ON QUADRATIC EQUATION PART 1, CLASS 10, CHAPTER 4, IT STARTS WITH THE SHAPE PARABOLA AND IT'S DAY TO DAY LIFE EXAMPLES, AS WE PROCEED FURTHER WE SOLVE SOME EXPRESSIONS, WE COVERT IT INTO QUADRATIC EQUATIONS. AFTERWARDS, WE LEARN HOW TO FORM STANDARD QUADRATIC EQUATIONS WITH EXAMPLES (WORD PROBLEMS).
Mathematics 9 Lesson 1-B: Solving Quadratic Equations using Quadratic FormulaJuan Miguel Palero
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using the Quadratic Formula. It also discusses the steps in solving quadratic equations using the method of Quadratic Formula.
Hello everyone! I hope this short powerpoint presentation can help you in understanding quadratic formula, especially for those students that are having hard time to cope up in this topic.
IT'S A PRESENTATION ON QUADRATIC EQUATION PART 1, CLASS 10, CHAPTER 4, IT STARTS WITH THE SHAPE PARABOLA AND IT'S DAY TO DAY LIFE EXAMPLES, AS WE PROCEED FURTHER WE SOLVE SOME EXPRESSIONS, WE COVERT IT INTO QUADRATIC EQUATIONS. AFTERWARDS, WE LEARN HOW TO FORM STANDARD QUADRATIC EQUATIONS WITH EXAMPLES (WORD PROBLEMS).
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