Additional solved examples for Quadratic Formula

Example 1:    x² + x - 2 = 0

Step 1: Comparing the given equation with

ax² + bx + c = 0

• a = 1
• b = 1
• c = -2

Step 2: Applying the Quadratic Formula,

Positive root

Negative root:

Answer: The roots are 1 and -2

Example 2: 2x² + √5x - 5 = 0

Step 1: Compare the equation with,

ax² + bx = c = 0

• a = 2
• b = √5
• c = -5

Step 2: Applying the Quadratic Formula,

(Positive root)

(Negative root)

Answer: Hence the roots are √5/2 and -√5

Example 3:

Step 1: Convert the given quadratic equation
to standard form

Multiplying throughout by 'x':	2x2 - 1 = 7x
Rewrite in standard form,	2x2 - 7x - 1 = 0

Step 2: Compare the given equation with

ax² + bx + c = 0

• a = 2
• b = -7
• c = -1

Step 3: Apply the Quadratic Formula

(Positive root)

(Negative root)

Answer:

Hence the roots are

and

Example 4:

(x + 1)/(x + 3) = (3x + 2)/(2x + 3)

Step 1: Convert the given quadratic equation
to standard form

By cross multiplication,	(x + 1)(2x + 3) = (3x + 2)(x + 3)
By FOIL method,	2x2 + 5x + 3 = 3x2 + 11x + 6
Simplify the equation,	(2x2 + 5x + 3) - (3x2 + 11x + 6) = 0
	-x2 - 6x - 3 = 0
Multiply throughout by -1,	x2 + 6x + 3 = 0

Step 2: Compare the given equation with

ax² + bx + c = 0

• a = 1
• b = 6
• c = 3

Step 3: Apply the Quadratic Formula

x = -3 +/- √6

Positive root: x = -3 + √6

Negative root: x = -3 - √6

Answer: Hence the roots are (-3 + √6)
and (-3 - √6).

Example 5:

√3x2 + 10x - 8√3

Step 1: Compare the given equation with

ax² + bx + c = 0

• a = √3
• b = 10
• c = -8√3

Step 2: Apply the Quadratic Formula

(Positive root)

(Negative root)

Answer:

Hence the two roots are (after ratinalizing)

(2√3)/3

and

-4√3