CBSE Class 10 Maths: Quadratic Equations Ex 4.1 - Concepts & Practice

Understanding Quadratic Equations

A quadratic equation in the variable 'x' is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are real numbers, and a ≠ 0. The most important characteristic to remember is that the highest power (or degree) of the variable in a quadratic equation must be 2. If 'a' were zero, the term ax² would vanish, and the equation would reduce to bx + c = 0, which is a linear equation, not quadratic. The terms 'a', 'b', and 'c' are coefficients, with 'a' being the coefficient of x², 'b' the coefficient of x, and 'c' the constant term. Understanding this standard form is the first step towards mastering quadratic equations. They appear in various real-life scenarios, from calculating projectile trajectories to optimizing areas, making their study incredibly valuable. You'll often encounter equations that don't immediately look like ax² + bx + c = 0; a key skill is to simplify and rearrange them into this standard form to confirm if they are indeed quadratic.

Step-by-Step: Identifying if an Equation is Quadratic

1. Step 1: Simplify Both Sides of the Equation — Expand all products, remove parentheses, and simplify any algebraic expressions on both the left-hand side (LHS) and the right-hand side (RHS) of the equation. Use algebraic identities like (a+b)² = a² + 2ab + b² or (a-b)² = a² - 2ab + b² where applicable.
2. Step 2: Transpose All Terms to One Side — Move all terms from the RHS to the LHS, making the RHS equal to zero. Remember to change the sign of terms when moving them across the equals sign. Combine like terms (terms with the same variable and power).
3. Step 3: Check the Highest Power of the Variable — After simplification and transposition, observe the resulting equation. If the highest power of the variable (usually 'x') is exactly 2, then the equation is a quadratic equation. If the highest power is 1 (linear) or any other value, it is not quadratic. Also, ensure the coefficient of x² (our 'a') is not zero.

Worked Examples: Forming Quadratic Equations

• Example 1: Check whether (x + 1)² = 2(x - 3) is a quadratic equation. A: Step 1: Expand both sides. LHS: (x + 1)² = x² + 2(x)(1) + 1² = x² + 2x + 1 RHS: 2(x - 3) = 2x - 6 So, the equation becomes x² + 2x + 1 = 2x - 6 Step 2: Transpose all terms to one side. x² + 2x + 1 - 2x + 6 = 0 Step 3: Simplify. x² + 7 = 0 Step 4: Check the highest power. The highest power of the variable 'x' is 2. The coefficient of x² is 1 (which is not 0). Final Answer: Yes, the given equation is a quadratic equation (in the form ax² + bx + c = 0, where a=1, b=0, c=7). Example 2: The area of a rectangular plot is 528 m². The length of the plot is one more than twice its breadth. We need to find the length and breadth of the plot (represent in quadratic equation form). A: Step 1: Define variables. Let the breadth of the rectangular plot be 'x' meters. Step 2: Express the length in terms of 'x'. The length is one more than twice its breadth, so length = (2x + 1) meters. Step 3: Use the given information (area). Area of rectangle = Length × Breadth 528 = (2x + 1) × x Step 4: Formulate the equation and convert to standard quadratic form. 528 = 2x² + x Transpose all terms to one side: 2x² + x - 528 = 0 Step 5: Verify if it is a quadratic equation. The highest power of 'x' is 2, and the coefficient of x² is 2 (not zero). Final Answer: The required quadratic equation representing the situation is 2x² + x - 528 = 0.

Exam Tip: Avoiding Common Mistakes in Ex 4.1

Students often make mistakes during algebraic expansion and simplification. Be very careful with signs, especially when multiplying negative numbers or transposing terms. A common error is simplifying (x+a)² incorrectly as x² + a² instead of x² + 2ax + a². For word problems, read the problem statement multiple times to accurately define variables and translate relationships into algebraic expressions. Sometimes, students forget to transpose all terms to one side and check the highest power; ensure the equation is in the ax² + bx + c = 0 format before concluding whether it's quadratic or not. Practice makes perfect – dedicate time to solving various types of problems from the NCERT textbook and reference books.

Practice Questions with Solutions

• Q: Check whether (x - 2)² + 1 = 2x - 3 is a quadratic equation. A: Step 1: Expand the LHS and simplify. (x - 2)² + 1 = (x² - 4x + 4) + 1 = x² - 4x + 5 So, the equation is x² - 4x + 5 = 2x - 3 Step 2: Transpose all terms to one side. x² - 4x + 5 - 2x + 3 = 0 Step 3: Combine like terms. x² - 6x + 8 = 0 Step 4: Check the highest power. The highest power of 'x' is 2. The coefficient of x² is 1 (not 0). Final Answer: Yes, it is a quadratic equation.
• Q: Check whether x(x + 1) + 8 = (x + 2)(x - 2) is a quadratic equation. A: Step 1: Expand both sides. LHS: x(x + 1) + 8 = x² + x + 8 RHS: (x + 2)(x - 2) = x² - 2² = x² - 4 (using (a+b)(a-b) = a²-b² identity) So, the equation is x² + x + 8 = x² - 4 Step 2: Transpose all terms to one side. x² + x + 8 - x² + 4 = 0 Step 3: Combine like terms. x + 12 = 0 Step 4: Check the highest power. The highest power of 'x' is 1 (not 2). Final Answer: No, it is not a quadratic equation; it is a linear equation.
• Q: Represent the following situation in the form of a quadratic equation: The product of two consecutive positive integers is 306. We need to find the integers. A: Step 1: Define variables for consecutive integers. Let the first positive integer be 'x'. Then, the next consecutive positive integer will be 'x + 1'. Step 2: Formulate the equation based on their product. According to the problem, their product is 306. x(x + 1) = 306 Step 3: Convert to standard quadratic form. x² + x = 306 x² + x - 306 = 0 Step 4: Verify. The highest power of 'x' is 2, and the coefficient of x² is 1 (not zero). Final Answer: The quadratic equation representing the situation is x² + x - 306 = 0.
• Q: Check whether (x + 2)³ = x³ - 4 is a quadratic equation. A: Step 1: Expand the LHS. (x + 2)³ = x³ + 3(x)²(2) + 3(x)(2)² + 2³ = x³ + 6x² + 12x + 8 So, the equation is x³ + 6x² + 12x + 8 = x³ - 4 Step 2: Transpose all terms to one side. x³ + 6x² + 12x + 8 - x³ + 4 = 0 Step 3: Combine like terms. 6x² + 12x + 12 = 0 Step 4: Check the highest power. The highest power of 'x' is 2. The coefficient of x² is 6 (not 0). Final Answer: Yes, it is a quadratic equation.
• Q: A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, it would have taken 3 hours more to cover the same distance. Represent this situation as a quadratic equation. A: Step 1: Define variables for speed and time. Let the uniform speed of the train be 'v' km/h. Time taken = Distance / Speed = 480/v hours. Step 2: Consider the new scenario. New speed = (v - 8) km/h. New time taken = 480/(v - 8) hours. Step 3: Relate the new time to the original time. The new time is 3 hours more than the original time: 480/(v - 8) = 480/v + 3 Step 4: Solve the equation to convert to standard quadratic form. Multiply by v(v-8) to clear denominators: 480v = 480(v - 8) + 3v(v - 8) 480v = 480v - 3840 + 3v² - 24v Transpose all terms to one side: 0 = 3v² - 24v - 3840 Alternatively, 3v² - 24v - 3840 = 0 Step 5: Simplify by dividing by 3 (optional, but good practice). v² - 8v - 1280 = 0 Final Answer: The quadratic equation representing the situation is v² - 8v - 1280 = 0.

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