Quadratic Equations

Quadratic equations are fundamental in algebra. They are polynomial equations of the second degree. The standard form of a quadratic equation is:

ax² + bx + c = 0

where 'a', 'b', and 'c' are coefficients, and 'x' is the variable. The coefficient 'a' cannot be zero (a ≠ 0). If a = 0, the equation becomes a linear equation. The solutions or roots of a quadratic equation are the values of 'x' that satisfy the equation. A quadratic equation can have at most two distinct real roots.

Methods to Solve Quadratic Equations

There are several methods to find the roots of a quadratic equation:

1. Factoring Method

This method involves rewriting the quadratic expression as a product of two linear factors.

Example: Solve x² + 5x + 6 = 0

We need to find two numbers that multiply to 6 (c) and add up to 5 (b). These numbers are 2 and 3.

So, we can rewrite the equation as:

x² + 2x + 3x + 6 = 0

Factor by grouping:

x(x + 2) + 3(x + 2) = 0

(x + 2)(x + 3) = 0

For the product to be zero, at least one of the factors must be zero.

x + 2 = 0 => x = -2

x + 3 = 0 => x = -3

The roots are -2 and -3.

2. Quadratic Formula Method

This is a universal method that works for all quadratic equations. The roots are given by the formula:

x = [-b ± √(b² - 4ac)] / 2a

The term b² - 4ac is called the discriminant (Δ). The nature of the roots depends on the value of the discriminant:

• If Δ > 0, there are two distinct real roots.
• If Δ = 0, there is exactly one real root (or two equal real roots).
• If Δ < 0, there are two complex roots (no real roots).

Example: Solve 2x² - 7x + 3 = 0

Here, a = 2, b = -7, c = 3.

First, calculate the discriminant:

Δ = b² - 4ac = (-7)² - 4(2)(3) = 49 - 24 = 25

Since Δ > 0, there are two distinct real roots.

Using the quadratic formula:

x = [-(-7) ± √25] / (2 * 2)

x = [7 ± 5] / 4

Two roots:

x₁ = (7 + 5) / 4 = 12 / 4 = 3

x₂ = (7 - 5) / 4 = 2 / 4 = 1/2

The roots are 3 and 1/2.

3. Completing the Square Method

This method involves manipulating the equation to form a perfect square trinomial.

Example: Solve x² + 6x + 5 = 0

Move the constant term to the right side:

x² + 6x = -5

Take half of the coefficient of x (which is 6/2 = 3) and square it (3² = 9). Add this value to both sides of the equation:

x² + 6x + 9 = -5 + 9

The left side is now a perfect square:

(x + 3)² = 4

Take the square root of both sides:

x + 3 = ±√4

x + 3 = ±2

Two possible values for x:

x + 3 = 2 => x = 2 - 3 = -1

x + 3 = -2 => x = -2 - 3 = -5

The roots are -1 and -5.

Comparison of Roots

In quantitative aptitude exams, you are often given two quadratic equations and asked to compare their roots. You need to solve both equations and then compare the values of x and y (if the equations are in terms of x and y).

Example:

Equation I: x² - 5x + 6 = 0

Equation II: y² - 7y + 12 = 0

Solving Equation I:

x² - 2x - 3x + 6 = 0

x(x - 2) - 3(x - 2) = 0

(x - 2)(x - 3) = 0

Roots for x are 2 and 3.

Solving Equation II:

y² - 3y - 4y + 12 = 0

y(y - 3) - 4(y - 3) = 0

(y - 3)(y - 4) = 0

Roots for y are 3 and 4.

Now compare the roots:

x values: {2, 3}

y values: {3, 4}

We see that:

• x = 2 is less than y = 3 and y = 4.
• x = 3 is equal to y = 3.
• x = 3 is less than y = 4.

Therefore, we can conclude that x ≤ y.

Percentage

Percentage is a way of expressing a number as a fraction of 100. The word "percent" means "per hundred". It is denoted by the symbol "%". For example, 50% means 50 out of 100, which can be written as 50/100 or 0.5. Percentages are widely used in finance, statistics, and everyday life to understand proportions and changes.

Converting Fractions and Decimals to Percentages

To convert a fraction or a decimal into a percentage, you multiply it by 100 and add the "%" sign.

Formula: Percentage = (Fraction or Decimal) × 100 %

Example 1: Convert the fraction 3/4 to a percentage.

Percentage = (3/4) × 100 % = (3 × 25) % = 75%

Example 2: Convert the decimal 0.65 to a percentage.

Percentage = 0.65 × 100 % = 65%

Converting Percentages to Fractions and Decimals

To convert a percentage to a fraction, divide the percentage value by 100 and remove the "%" sign. Simplify the fraction if possible.

Formula: Fraction = Percentage / 100

Example 1: Convert 40% to a fraction.

Fraction = 40 / 100 = 4/10 = 2/5

To convert a percentage to a decimal, divide the percentage value by 100 and remove the "%" sign.

Formula: Decimal = Percentage / 100

Example 2: Convert 75% to a decimal.

Decimal = 75 / 100 = 0.75

Calculating Percentage of a Number

To find the percentage of a given number, you convert the percentage to a decimal or a fraction and then multiply it by the number.

Formula: Percentage of a Number = (Percentage / 100) × Number

Example 1: Calculate 20% of 150.

(20 / 100) × 150 = 0.20 × 150 = 30

Example 2: Calculate 12.5% of 800.

12.5% = 12.5 / 100 = 0.125. Alternatively, 12.5% = 1/8.

Using decimal: 0.125 × 800 = 100

Using fraction: (1/8) × 800 = 100

Percentage Increase and Decrease

Percentage Increase: When a quantity increases, the percentage increase is calculated with respect to the original value.

Formula: Percentage Increase = [(New Value - Original Value) / Original Value] × 100 %

Example: If the price of a pen increased from ₹10 to ₹12, find the percentage increase.

Percentage Increase = [(12 - 10) / 10] × 100 % = (2 / 10) × 100 % = 0.2 × 100 % = 20%

Percentage Decrease: When a quantity decreases, the percentage decrease is calculated with respect to the original value.

Formula: Percentage Decrease = [(Original Value - New Value) / Original Value] × 100 %

Example: If the number of students in a class decreased from 50 to 40, find the percentage decrease.

Percentage Decrease = [(50 - 40) / 50] × 100 % = (10 / 50) × 100 % = 0.2 × 100 % = 20%

Successive Percentage Changes

When a quantity undergoes two or more percentage changes one after another, the final change is calculated based on the value after the previous change. The formula for two successive percentage changes is:

Formula: Net Percentage Change = [ A + B + (AB / 100) ] %

Where A and B are the percentage changes. If the change is an increase, the value is positive; if it's a decrease, the value is negative.

Example: A price is increased by 10% and then decreased by 20%. Find the net percentage change.

Here, A = +10% (increase) and B = -20% (decrease).

Net Percentage Change = [ 10 + (-20) + (10 × -20 / 100) ] %

= [ 10 - 20 + (-200 / 100) ] %

= [ -10 - 2 ] % = -12%

The net change is a 12% decrease.

Percentage Word Problems

Percentage problems often appear in word format. The key is to identify the original value, the new value, and what is being asked (percentage increase/decrease, a specific percentage of a number, etc.). Always read the question carefully.

Example: A shopkeeper sells an article for ₹200, making a profit of 25%. What is the cost price of the article?

Here, the selling price (SP) is ₹200. The profit is 25%. This means the profit is 25% of the Cost Price (CP).

SP = CP + Profit

SP = CP + (25% of CP)

SP = CP + (0.25 × CP)

SP = 1.25 × CP

₹200 = 1.25 × CP

CP = ₹200 / 1.25 = ₹200 / (5/4) = ₹200 × (4/5) = ₹160.

The cost price is ₹160.

Profit and Loss

Profit and Loss are concepts related to the financial transaction of buying and selling goods or services. They help determine the financial outcome of a business activity.

Key Terms

• Cost Price (CP): The price at which an article is purchased or manufactured.
• Selling Price (SP): The price at which an article is sold.
• Profit: Occurs when the Selling Price is greater than the Cost Price (SP > CP).
• Loss: Occurs when the Selling Price is less than the Cost Price (SP < CP).
• Overhead Charges: Expenses incurred in addition to the cost price, such as transportation, repair, installation, etc. These are added to the CP to get the total cost price.

Formulas for Profit

If SP > CP:

• Profit Amount = SP - CP
• Profit Percentage = (Profit Amount / CP) × 100 %

Example: A shopkeeper buys a toy for ₹80 and sells it for ₹100.

CP = ₹80, SP = ₹100.

Profit Amount = ₹100 - ₹80 = ₹20.

Profit Percentage = (₹20 / ₹80) × 100 % = (1/4) × 100 % = 25%.

Formulas for Loss

If SP < CP:

• Loss Amount = CP - SP
• Loss Percentage = (Loss Amount / CP) × 100 %

Example: A shopkeeper buys a book for ₹150 and sells it for ₹120.

CP = ₹150, SP = ₹120.

Loss Amount = ₹150 - ₹120 = ₹30.

Loss Percentage = (₹30 / ₹150) × 100 % = (1/5) × 100 % = 20%.

Calculating SP and CP when Percentage is Given

When the profit or loss percentage is given, we can calculate the SP if CP is known, or CP if SP is known.

1. To find SP when CP and Profit % are given:

SP = CP × (100 + Profit %) / 100

SP = CP × (100 + P) / 100

Example: Find the SP if CP = ₹500 and Profit = 20%.

SP = 500 × (100 + 20) / 100 = 500 × (120 / 100) = 500 × 1.2 = ₹600.

2. To find SP when CP and Loss % are given:

SP = CP × (100 - Loss %) / 100

SP = CP × (100 - L) / 100

Example: Find the SP if CP = ₹500 and Loss = 20%.

SP = 500 × (100 - 20) / 100 = 500 × (80 / 100) = 500 × 0.8 = ₹400.

3. To find CP when SP and Profit % are given:

CP = SP × 100 / (100 + Profit %)

CP = SP × 100 / (100 + P)

Example: Find the CP if SP = ₹600 and Profit = 20%.

CP = 600 × 100 / (100 + 20) = 600 × 100 / 120 = 600 × (10/12) = 50 × 10 = ₹500.

4. To find CP when SP and Loss % are given:

CP = SP × 100 / (100 - Loss %)

CP = SP × 100 / (100 - L)

Example: Find the CP if SP = ₹400 and Loss = 20%.

CP = 400 × 100 / (100 - 20) = 400 × 100 / 80 = 400 × (10/8) = 50 × 10 = ₹500.

Dishonest Dealers

These problems involve a seller who uses unfair means to increase their profit. Common tricks include using a faulty scale (selling less than the marked weight) or marking up prices and then offering discounts.

Scenario: A dealer claims to sell goods at CP but uses a faulty weight of 'x' grams instead of 1000 grams.

The dealer intends to sell 1000 grams (1 kg) but actually sells 'x' grams.

The cost price is for 1000 grams.

The selling price is calculated as if it were for 1000 grams, but the weight delivered is 'x' grams.

Profit % = [(True Weight - Faulty Weight) / Faulty Weight] × 100 %

Example: A dishonest dealer sells sugar at CP but uses a weight of 900 grams for every 1 kg (1000 grams) sold. Find his profit percentage.

Profit % = [(1000 - 900) / 900] × 100 % = (100 / 900) × 100 % = (1/9) × 100 % = 11 1/9 %.

Scenario: A dealer marks up the price of an article by a certain percentage and then offers a discount.

First, calculate the Marked Price (MP) after the markup.

MP = CP × (100 + Markup %) / 100

Then, calculate the SP after the discount.

SP = MP × (100 - Discount %) / 100

Finally, calculate the profit or loss.

Profit/Loss = SP - CP

Profit/Loss % = [(SP - CP) / CP] × 100 %

Example: A shopkeeper marks his goods 40% above the CP and then offers a 20% discount. Find the profit percentage.

Let CP = ₹100.

Markup = 40%.

MP = 100 × (100 + 40) / 100 = 100 × 140 / 100 = ₹140.

Discount = 20%.

SP = MP × (100 - 20) / 100 = 140 × (80 / 100) = 140 × 0.8 = ₹112.

Profit = SP - CP = 112 - 100 = ₹12.

Profit % = (12 / 100) × 100 % = 12%.

Partnership Problems (Related to Profit)

When two or more people invest in a business, the profit is usually shared in the ratio of their investments. If the duration of investment differs, the profit is shared in the ratio of (Investment × Time).

Example: A and B invest ₹10,000 and ₹15,000 respectively in a business. The total profit is ₹25,000. How is the profit shared?

Ratio of investments = 10,000 : 15,000 = 10 : 15 = 2 : 3.

Total parts = 2 + 3 = 5.

A's share of profit = (2/5) × 25,000 = ₹10,000.

B's share of profit = (3/5) × 25,000 = ₹15,000.

Example with Time: A invests ₹50,000 for 6 months, and B invests ₹60,000 for 4 months. The profit is ₹38,000. Find the share of each.

Ratio of investment × time = (50,000 × 6) : (60,000 × 4)

= 300,000 : 240,000

= 30 : 24

= 5 : 4 (Dividing by 6)

Total parts = 5 + 4 = 9.

A's share = (5/9) × 38,000 = ₹190,000 / 9 ≈ ₹21,111.11

B's share = (4/9) × 38,000 = ₹152,000 / 9 ≈ ₹16,888.89

Note: In exams, these usually work out to neat integer values.