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

1)
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 xMultiply 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 = 8^{2} - 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 = 0Identify 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)
## More References and LinksSolve Equations, Systems of Equations and InequalitiesProof of the Quadratic Formulas and Questions. Step by Step Quadratic Equation Solver. |