Solutions to Questions on Quadratic Equations

Detailed solutions and explanations to the questions in Solve Quadratic Equations Using Discriminants

1)
a)
Given
x 2 - 3 x + 2 = 0
Identify the coefficients a,b and c.
a = 1, b = - 3 , c = 2
Find the the discriminant
D = b
2 - 4 a c = (-3) 2 - 4 (1) (2) = 1.
Since the discriminant is positive, the above equation has two real solutions
x
1 = (-b + √D) / (2 a) = (3 + 1) / 2 = 2
x
2 = (- b - √D) / (2 a) = (3 - 1) / 2 = 1

b)
Given
x 2 / 2 = - 8 - 4 x
Multiply all terms in the above equation by 2, to eliminate the denominator, simplify and write it in standard form.
x
2 + 8 x + 16 = 0
Identify the coefficients a,b and c.
a = 1, b = 8 , c = 16
Find the discriminant D
D = b
2 - 4 a c = 82 - 4 (1)(16) = 64 - 64 = 0.
Since the discriminant is equal to zero, the above equation has one real solution.
x = - b / 2 a = -8/2 = -4

c)
Given
x 2 - 4x + 5 = 0
Identify the coefficients a,b and c.
a = 1, b = - 4 , c = 5
Find the discriminant D.
D = b
2 - 4 a c = (-4)2 - 4 (1)(5) = 16 - 20 = -4.
Since the discriminant is negative, the above equation has two complex solutions.
x = (- b + √ D ) /(2 a) = (4 + √(-4) ) / 2 = (4 + 2 i) / 2 = 2 + i
x = (- b - √ D ) /(2 a) = (4 - √(-4) ) / 2 = (4 - 2 i) / 2 = 2 - i
where i = √(-1) is the imaginary unit.

2)
Given
x 2 + x + m + 1 = 0
Identify the coefficients a,b and c.
a = 1, b = 1 , c = m + 1
Find the discriminant D
D = b2 - 4ac = 12 - 4(1)(m + 1) = -3 - 4m

a) For the equation to have one solution, the discriminant has to be equal to zero.
-3 - 4m = 0
Solve the above equation for m.
m = -3/4

b) For the equation to have 2 real solutions, the discriminant has to be greater than zero.
-3 - 4m > 0
The inequality -3 - 4m > 0 has the following solution set.
(-∞ , - 3/4)

c) For the equation to have 2 complex solutions, the discriminant has to be less than zero.
-3 - 4m < 0
The inequality -3 - 4m > 0 has the following solution set.
(- 3/4 , + ∞)

More References and Links

Questions on Quadratic Equations and their detailed solutions.
Solve Equations, Systems of Equations and Inequalities
Proof of the Quadratic Formulas and Questions.
Step by Step Quadratic Equation Solver.

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