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Q1(iv):
Find the roots of the following quadratic equations by factorisation: (iv) $2x^2 – x + \frac{1}{8} = 0$
Solution :
Given: The quadratic equation $2x^2 - x + \frac{1}{8} = 0$.
To find: The roots of the given quadratic equation by the method of factorisation.
Step 1: Simplifying the Equation
To make the factorisation process easier, we first eliminate the fraction by multiplying the entire equation by $8$.
$8 \times (2x^2 - x + \frac{1}{8}) = 8 \times 0$
$16x^2 - 8x + 1 = 0$
Step 2: Splitting the Middle Term
We need to factorise the quadratic expression $16x^2 - 8x + 1$. We look for two numbers such that:
1. Their product is equal to the product of the coefficient of $x^2$ and the constant term: $16 \times 1 = 16$.
2. Their sum is equal to the coefficient of $x$: $-8$.
The two numbers that satisfy these conditions are $-4$ and $-4$, since $(-4) \times (-4) = 16$ and $(-4) + (-4) = -8$.
Step 3: Factorising by Grouping
Rewrite the middle term $-8x$ as $-4x - 4x$:
$16x^2 - 4x - 4x + 1 = 0$
Group the terms into two pairs:
$(16x^2 - 4x) - (4x - 1) = 0$
Factor out the common terms from each group:
$4x(4x - 1) - 1(4x - 1) = 0$
Step 4: Extracting the Factors
Now, factor out the common binomial $(4x - 1)$:
$(4x - 1)(4x - 1) = 0$
$(4x - 1)^2 = 0$
Step 5: Finding the Roots
To find the roots, set each factor equal to zero [By the Zero Product Property]:
$4x - 1 = 0$
$4x = 1$
$x = \frac{1}{4}$
Since both factors are identical, the equation has two equal real roots.
Final Answer: The roots of the quadratic equation are $x = \frac{1}{4}$ and $x = \frac{1}{4}$.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.2
- Q1(i): Find the roots of the following quadratic equations by factorisation: (i) $x^2 – 3x – 10 = 0$
- Q1(ii): Find the roots of the following quadratic equations by factorisation: (ii) $2x^2 + x – 6 = 0$
- Q1(iii): Find the roots of the following quadratic equations by factorisation: (iii) $\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0$
- Q1(v): Find the roots of the following quadratic equations by factorisation: (v) $100x^2 – 20x + 1 = 0$
- Q2: Solve the problems given below. Represent the following situations mathematically: (i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with. (ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs.750. We would like to find out the number of toys produced on that day.
- Q3: Find two numbers whose sum is $27$ and product is $182$.
- Q4: Find two consecutive positive integers, sum of whose squares is $365$.
- Q5: The altitude of a right triangle is $7$ cm less than its base. If the hypotenuse is $13$ cm, find the other two sides.
- Q6: A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was $3$ more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ $90$, find the number of articles produced and the cost of each article.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
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