Mastering quadratic equations is a crucial milestone in your academic journey, especially as you prepare for your board exams. These algebraic expressions appear frequently in the board question papers, carrying substantial weightage that can significantly boost your overall percentage. Having accurate, step-by-step guidance is essential to avoid common calculation errors and to present your answers in a way that fetches maximum marks from examiners.
Finding reliable NCERT solutions can sometimes feel overwhelming when you are stuck on a tricky word problem or struggling to understand completing the square method. Whether you want to Browse all chapter notes for quick revisions or test your concepts using our Practice tests, having the right support system makes all the difference. Let's dive deep into comprehensive solutions for Class 10 Maths Chapter 4 to help you ace your CBSE exams.
Understanding Quadratic Equations in Class 10
Before jumping into the exercises, let's understand what a quadratic equation actually is. As per the guidelines provided on the official CBSE website, a second-degree polynomial equation in a single variable is known as a quadratic equation. Its standard form is given by:
$ax^2 + bx + c = 0$
where $a, b,$ and $c$ are real numbers, and $a \neq 0$.
To strengthen your fundamental understanding before tackling the exercises, feel free to Explore CircleOfLearning for additional interactive study materials. Let us now move on to solving the core textbook problems step by step.
Exercise Solutions: Quadratic Equations
Solving problems methodically is the key to mastering mathematics. Below, we break down key questions from the chapter with clear working, correct formulas, and highlighted final answers.
Checking for Quadratic Equations
Question 1: Check whether the following are quadratic equations: $(x + 1)^2 = 2(x - 3)$
Step 1: Expand the left-hand side using the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$.
$x^2 + 2x + 1 = 2x - 6$
Step 2: Bring all terms to the left-hand side to simplify the expression.
$x^2 + 2x + 1 - 2x + 6 = 0$
$x^2 + 7 = 0$
Final Answer: Yes, it is a quadratic equation because it can be written in the form $ax^2 + bx + c = 0$ where $a = 1$, $b = 0$, and $c = 7$.
Solving by Factorisation Method
Question 2: Find the roots of the quadratic equation $2x^2 - x + \frac{1}{8} = 0$
Step 1: Multiply the entire equation by 8 to clear the fraction and simplify calculations.
$8 \left(2x^2 - x + \frac{1}{8}\right) = 8 \times 0$
$16x^2 - 8x + 1 = 0$
Step 2: Split the middle term ($-8x$) into two parts whose product is $(16 \times 1 = 16)$ and sum is $-8$. These numbers are $-4$ and $-4$.
$16x^2 - 4x - 4x + 1 = 0$
Step 3: Factor by grouping the terms.
$4x(4x - 1) - 1(4x - 1) = 0$
$(4x - 1)(4x - 1) = 0$
Final Answer: The roots are $x = \frac{1}{4}$ and $x = \frac{1}{4}$.
Finding Roots Using the Quadratic Formula
Question 3: Find the roots of $2x^2 - 7x + 3 = 0$ using the quadratic formula.
Step 1: Compare the given equation with $ax^2 + bx + c = 0$ to identify the coefficients as outlined in the curriculum published by the official NCERT portal.
$a = 2, \quad b = -7, \quad c = 3$
Step 2: Calculate the discriminant ($D$) using the formula $D = b^2 - 4ac$.
$D = (-7)^2 - 4(2)(3)$
$D = 49 - 24 = 25$
Since $D > 0$, real and distinct roots exist.
Step 3: Apply the quadratic formula $x = \frac{-b \pm \sqrt{D}}{2a}$.
$x = \frac{-(-7) \pm \sqrt{25}}{2(2)}$
$x = \frac{7 \pm 5}{4}$
Step 4: Calculate both possible values for $x$.
- For addition: $x = \frac{7 + 5}{4} = \frac{12}{4} = 3$
- For subtraction: $x = \frac{7 - 5}{4} = \frac{2}{4} = \frac{1}{2}$
Final Answer: The roots of the equation are $x = 3$ and $x = \frac{1}{2}$.
Word Problems on Quadratic Equations
Word problems often intimidate students, but translating sentences into mathematical equations becomes easy with consistent practice.
Question 4: Find two consecutive positive integers, sum of whose squares is 365.
Step 1: Let the first positive integer be $x$. Therefore, the next consecutive positive integer is $(x + 1)$.
Step 2: Set up the equation according to the given condition.
$x^2 + (x + 1)^2 = 365$
Step 3: Expand and simplify the equation.
$x^2 + x^2 + 2x + 1 = 365$
$2x^2 + 2x + 1 - 365 = 0$
$2x^2 + 2x - 364 = 0$
Divide the entire equation by 2:
$x^2 + x - 182 = 0$
Step 4: Factorise the quadratic equation by splitting the middle term. Find two numbers whose product is $-182$ and sum is $1$. These numbers are $14$ and $-13$.
$x^2 + 14x - 13x - 182 = 0$
$x(x + 14) - 13(x + 14) = 0$
$(x - 13)(x + 14) = 0$
This gives $x = 13$ or $x = -14$. Since the problem specifies positive integers, we discard $-14$.
Final Answer: The integers are $13$ and $13 + 1 = 14$.
Tips to Score Full Marks in Board Exams
Scoring a century in Class 10 Maths is entirely achievable with the right strategy. Keep these expert tips in mind while answering board questions:
- Always write down the given parameters: Start your solutions by clearly stating what values of $a, b,$ and $c$ you are using.
- Show intermediate calculation steps: Examiners award step-marking. Even if your final calculation slips, correct formulas and setup will earn you partial credit.
- Verify your answers: Substitute your derived roots back into the original equation during your reading time to check if they satisfy $LHS = RHS$.
- Include units in word problems: If a question involves speed, age, or distance, never forget to write the proper units (like cm, years, or km/h) in your final answer statement.
- Practice previous year questions: Solve past board papers to get familiar with recurring patterns and time management during the exam.
Frequently Asked Questions
What is a quadratic equation in Class 10 Maths?
A quadratic equation is a second-degree polynomial equation represented in the standard form $ax^2 + bx + c = 0$, where $a \neq 0$ and $a, b, c$ are real numbers.
What are the main methods to solve quadratic equations?
The three primary methods taught in Class 10 are factorization (splitting the middle term), completing the square method, and using the quadratic formula (Sridharacharya formula).
What is the discriminant formula and why is it important?
The discriminant is given by $D = b^2 - 4ac$. It tells us the nature of the roots: if $D > 0$ (two distinct real roots), $D = 0$ (two equal real roots), and $D < 0$ (no real roots).
Are these NCERT solutions updated for the latest CBSE syllabus?
Yes, all the solutions provided here strictly align with the rationalized CBSE syllabus and NCERT textbook guidelines for the current academic session.
How can I improve my speed in solving quadratic word problems?
Practice translating statements into mathematical expressions daily. Break down long sentences into smaller relationships, assign variables carefully, and verify your equations before solving.
Conclusion
Mastering quadratic equations requires a solid grasp of fundamental formulas, regular practice of word problems, and neat presentation during exams. By following structured steps and verifying your work, you can easily tackle even the most challenging questions thrown your way by examiners.
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