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Q1(i):
Find the roots of the following quadratic equations by factorisation: (i) $x^2 – 3x – 10 = 0$
Solution :
Given: A quadratic equation $x^2 - 3x - 10 = 0$.
To Find: The roots of the given quadratic equation using the method of factorisation.
Step 1: Understanding the Factorisation Method (Splitting the Middle Term)
To factorise a quadratic equation of the form $ax^2 + bx + c = 0$, we look for two numbers, say $p$ and $q$, such that:
1. The product of the numbers is equal to the product of the coefficient of $x^2$ and the constant term: $p \times q = a \times c$.
2. The sum of the numbers is equal to the coefficient of $x$: $p + q = b$.
In the given equation $x^2 - 3x - 10 = 0$, we have $a = 1$, $b = -3$, and $c = -10$.
Therefore, we need two numbers such that:
Product ($p \times q$) = $1 \times (-10) = -10$
Sum ($p + q$) = $-3$
Step 2: Identifying the Factors
We list the pairs of factors of $-10$:
- $(1, -10) \implies 1 + (-10) = -9$
- $(-1, 10) \implies -1 + 10 = 9$
- $(2, -5) \implies 2 + (-5) = -3$
- $(-2, 5) \implies -2 + 5 = 3$
The pair that satisfies both conditions is $2$ and $-5$.
Step 3: Splitting the Middle Term
We rewrite the middle term $-3x$ as $2x - 5x$:
$x^2 + 2x - 5x - 10 = 0$
Step 4: Grouping and Factoring
Group the terms into two pairs:
$(x^2 + 2x) - (5x + 10) = 0$
Factor out the common terms from each group:
$x(x + 2) - 5(x + 2) = 0$
Now, factor out the common binomial $(x + 2)$:
$(x + 2)(x - 5) = 0$
Step 5: Finding the Roots
According to the Zero Product Property, if the product of two factors is zero, then at least one of the factors must be zero.
Case 1: $x + 2 = 0$
$x = -2$
Case 2: $x - 5 = 0$
$x = 5$
Final Answer: The roots of the quadratic equation $x^2 - 3x - 10 = 0$ are $x = -2$ and $x = 5$.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.2
- 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(iv): Find the roots of the following quadratic equations by factorisation: (iv) $2x^2 – x + \frac{1}{8} = 0$
- 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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