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

2. Quadratic Equations
In One Variable

3. Quadratic Equation
an equation of the form
ax2 + bx + c = 0
where a, b, and c are real numbers.

4. Types of Quadratic Equations
 Complete Quadratic
3x2 + 5x + 6 = 0
 Incomplete/Pure Quadratic Equation
3x2 - 6 = 0

5. Solving an
Incomplete Quadratic

6. Example 1. Solve: x2 – 4 = 0
Solution:
x2 – 4 = 0
x2 = 4
√x² = √4
x = ± 2

7. Example 2. Solve: 5x² - 11 = 49
Solution:
5x² - 11 = 49
5x² = 49 + 11
5x² = 60
x² = 12
x = ±√12
x = ±2√3

8. Solving
Quadratic Equation

9. By Factoring
1. Place all terms in the left member of the equation, so that
the right member is zero.
2. Factor the left member.
3. Set each factor that contains the unknown equal to zero.
4. Solve each of the simple equations thus formed.
5. Check the answers by substituting them in the original
equation.

10. 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

11. By Completing the Square
1. Write the equation with the variable terms in the left member and
the constant term in the right member.
2. 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.
3. Take one-half the coefficient of x, square this quantity, and add the
result to both members.
4. Find the square root of both members, placing a ± sign before the
square root of the right member.
5. Solve the resulting equation for x.

12. Example: x² - 8x + 7 = 0


13. By Quadratic Formula
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎

14. Example: 3x² - 2x - 7 = 0
Where,
a = 3 b = -2 c = -7
x =
−𝑏± 𝑏2−4𝑎𝑐
2𝑎
X =
2 ± 88
6
X =
− −2 ± (−2)2−4(3)(−7)
2(3)
X =
2 ± 2 22
6
X =
2 ± 4+84
6
X =
1 ± 22
3

15. 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

16. 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