Quadratic Equations Mr D. Quiz 1 Reviewer

​Quadratic Equations and Inequalities

By Mr. D

Standard Form of a Quadratic Equation:

ax2+bx+c=0

Quadratic Formula:

x = (-b ± √(b² - 4ac)) / (2a)

Discriminant:

D = b² - 4ac

Sample Problem 1

Find the discriminant (D) of the quadratic equation:

x² - 6x + 9 = 0

Answer:
D = (-6)² - 4(1)(9)

D = 36 - 36

D = 0

Sample Problem 2

Solve for x in the quadratic equation:

2x² - 4x - 6 = 0

Answer:
x = ( -(-4) ± √((-4)² - 4(2)(-6)) ) / (2(2))

x = ( 4 ± √(16 + 48) ) / 4

x = ( 4 ± √64 ) / 4

x = (4 + 8) / 4 = 12 / 4 = 3

x = (4 - 8) / 4 = -4 / 4 = -1

Sample Problem 3

The quadratic equation is:

Ax² - 4x + 3 = 0

It is known that the discriminant D = 16. Find the value of A.

Answer:
D=b2−4ac
16 = (-4)² - 4(A)(3)
16 - 12A = 16

-12A = 16 - 16

-12A = 0

A = 0

Sample Problem 4

A quadratic equation has roots x = 4 and x = -6, and the constant term c = 25. Find the values of a and b in the standard quadratic equation ax² + bx + c = 0.

Answer:
x₁ × x₂ = c / a

4 × (-6) = 25 / a

-24 = 25 / a

a = -25 / 24

4 + (-6) = -b / (-25 / 24)

-2 = b / (25 / 24)

b = -2 × (25 / 24)

b = -50 / 24

b = -25 / 12

Sample Problem 5

Solve the equation, simplify it into the quadratic form:

3x² + 5x - 2 = x² + 4x + 6

Answer:
3x² + 5x - 2 - x² - 4x - 6 = 0

(3x² - x²) + (5x - 4x) + (-2 - 6) = 0

2x² + x - 8 = 0