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Q1(ii):
Find the roots of the following quadratic equations by factorisation: (ii) $2x^2 + x – 6 = 0$
Solution :
Given: A quadratic equation $2x^2 + x - 6 = 0$.
To find: The roots of the given quadratic equation using the method of factorisation (splitting the middle term).
Step 1: Identify the coefficients of the quadratic equation.
The standard form of a quadratic equation is $ax^2 + bx + c = 0$.
Comparing $2x^2 + x - 6 = 0$ with the standard form:
$a = 2$
$b = 1$
$c = -6$
Step 2: Determine the product and the sum for splitting the middle term.
To factorise by splitting the middle term, we need to find two numbers such that:
1. Their product is equal to $a \times c = 2 \times (-6) = -12$.
2. Their sum is equal to $b = 1$.
Step 3: Find the two numbers.
We look for factors of $-12$ that add up to $1$:
Factors of $-12$: $(-1, 12), (1, -12), (-2, 6), (2, -6), (-3, 4), (3, -4)$.
Checking the sums:
$-3 + 4 = 1$.
Thus, the two numbers are $4$ and $-3$.
Step 4: Rewrite the middle term and factorise by grouping.
Rewrite the equation $2x^2 + x - 6 = 0$ as:
$2x^2 + 4x - 3x - 6 = 0$ [Splitting the middle term $x$ into $4x - 3x$]
Group the terms:
$(2x^2 + 4x) - (3x + 6) = 0$
Factor out the common terms from each group:
$2x(x + 2) - 3(x + 2) = 0$ [Factoring out $2x$ from the first group and $3$ from the second group]
$(2x - 3)(x + 2) = 0$ [Taking $(x + 2)$ as a common factor]
Step 5: Find the roots by setting each factor to zero.
According to the Zero Product Property, if the product of two factors is zero, at least one of the factors must be zero.
Case 1: $2x - 3 = 0$
$2x = 3$
$x = \frac{3}{2}$
Case 2: $x + 2 = 0$
$x = -2$
Final Answer: The roots of the quadratic equation $2x^2 + x - 6 = 0$ are $x = \frac{3}{2}$ and $x = -2$.
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(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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