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The standard form of the quadratic equation is ax² + bx + c = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
where a, b are coefficients of x², x and c are constant.

Example 1.
Solve the equation x² – 9 = 0

Solution:

Given,
x² – 9 = 0
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 1
b = 0
c = -9
x = [-0 ± √(0² – 4.1.(-9))]/2.1
x = ±6i/2
x = +3i
x = -3i

Example 2.
Solve the equation 2x² + 5x = 9

Solution:

Given,
2x² + 5x = 9
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
2x² + 5x – 9 = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 2
b = 5
c = -9
x = [-5 ± √(5² – 4.2.(-9))]/2.2
x = [-5 ± √(25 + 72)]/4
x = [-5 ± √97]/4
x = [-5 + √97]/4
x = [-5 – √97]/4

Example 3.
Solve the equation 6x² – 11 = 0

Solution:

Given,
6x² – 11 = 0
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 6
b = 0
c = -11
x = [-0 ± √(0² – 4.6.(-11))]/2.6
x = ±√264/12
x = +√264/12 = 1.35401
x = -√264/12 = -1.35401

Example 4.
Solve the equation n² + 10n + 21 = 0

Solution:

Given,
n² + 10n + 21 = 0
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 1
b = 10
c = 21
x = [-10 ± √(10² – 4.1.(21))]/2.1
x = [-10 ± √(100 – 84)]/2
x = [-10 ± √(16)]/2
x = [-10 ± 4]/2
x = [-10 + 4]/2
x = -6/2
x = -3
x = [-10 – 4]/2
x = -14/2
x = -7

Example 5.
Solve the equation 6x² + 12x + 25 = 0

Solution:

Given,
6x² + 12x + 25 = 0
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 6
b = 12
c = 25
x = [-12 ± √(12² – 4.6.(25))]/2.6
x = [-12 ± √(144 – 600)]/12
x = [-12 ± √(-456)]/12
x = [-12 ±2√(114)i]/12
x = -1 + 1.779i
x = -1 – 1.779i

Example 6.
Solve the equation x² – 3x + 10x + 7 = 0

Solution:

Given,
x² – 3x + 10x + 7 = 0
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
x² – 3x + 10x + 7 = 0
x² + 7x + 7 = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 1
b = 7
c = 7
x = [-7 ± √(7² – 4.1.(7))]/2.1
x = [-7 ± √(21)]/2
x = [-7 + √(21)]/2
x = [-7 – √(21)]/2

Example 7.
Solve the equation 6x² + 2x = -3

Solution:

Given,
6x² + 2x = -3
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
6x² + 2x + 3 = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 6
b = 2
c = 3
x = [-2 ± √(2² – 4.6.(3))]/2.6
x = [-2 ± √(4 – 72)]/12
x = [-7 ± √(-68)]/12
x = [-7 ± √(21)]/12
x = [-7 + √(21)]/12
x = [-7 – √(21)]/12

Example 8.
Solve the equation 3x² – 5x = 8

Solution:

Given,
3x² – 5x = 8
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
3x² – 5x – 8 = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 3
b = -5
c = -8
x = [-(-5) ± √(5² – 4.3.(-8))]/2.3
x = [5 ± √(25 – (-96))]/6
x = [5 ± √(-68)]/6
x = [5 ± √(121)]/6
x = [5 ± 11]/6
x = [5 + 11]/6
x = 16/6 = 8/3
x = [5 – 11]/6
x = -6/6
x = -1

Example 9.
Solve the equation x² + 2x + 1 = 0

Solution:

Given,
x² + 2x + 1 = 0
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
x² + 2x + 1 = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 1
b = 2
c = 1
x = [-2 ± √(2² – 4.1.(1))]/2.1
x = [-2 ± √(4 – 4)]/2
x = [-2 ± √(0)]/2
x = [-2± 0]/2
x = -2/2
x = -1

Example 10.
Solve the equation 3x² – 23 = 6x

Solution:

Given,
3x² – 23 = 6x
Convert the given equation into a standard form of the quadratic equation.
The standard form of the quadratic equation is ax² + bx + c = 0
3x² – 6x – 23 = 0
(α, β) = [-b ± √(b² – 4ac)]/2a
a = 3
b = -6
c = -23
x = [-(-6) ± √((-6)² – 4.3.(-23))]/2.3
x = [6 ± √(36 – (-276))]/6
x = [6 ± 2√(78)]/6
x = [6 + 2√(78)]/6
x = [6 – 2√(78)]/6