Find the best tutors and institutes for Class 10 Tuition
Q2(ii):
Represent the following situations in the form of quadratic equations : (ii) The product of two consecutive positive integers is $306$. We need to find the integers.
Solution :
Given: The product of two consecutive positive integers is $306$.
To find: Represent the given situation in the form of a quadratic equation.
Step 1: Defining the variables
Let the first positive integer be $x$.
Since the integers are consecutive, the next integer must be $x + 1$.
[Assumption: $x$ is a positive integer, therefore $x > 0$].
Step 2: Formulating the equation based on the given condition
According to the problem, the product of these two consecutive integers is $306$.
Mathematically, this is expressed as:
$x(x + 1) = 306$
Step 3: Expanding the expression
Distribute $x$ into the parentheses:
$x^2 + x = 306$
Step 4: Rearranging into standard quadratic form
A quadratic equation is represented in the standard form $ax^2 + bx + c = 0$.
Subtract $306$ from both sides of the equation to set the right side to zero:
$x^2 + x - 306 = 0$
Step 5: Verification of the form
Comparing $x^2 + x - 306 = 0$ with the standard form $ax^2 + bx + c = 0$:
Here, $a = 1$, $b = 1$, and $c = -306$.
Since the highest power of the variable $x$ is $2$, this is a quadratic equation.
Final Answer: The quadratic equation representing the given situation is $x^2 + x - 306 = 0$, where $x$ is the smaller integer.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.1
- Q1(i): Check whether the following are quadratic equations : (i) $(x + 1)^2 = 2(x – 3)$
- Q1(ii): Check whether the following are quadratic equations : (ii) $x^2 – 2x = (–2)(3 – x)$
- Q1(iii): Check whether the following are quadratic equations : (iii) $(x – 2)(x + 1) = (x – 1)(x + 3)$
- Q1(iv): Check whether the following are quadratic equations : (iv) $(x – 3)(2x +1) = x(x + 5)$
- Q1(v): Check whether the following are quadratic equations : (v) $(2x – 1)(x – 3) = (x + 5)(x – 1)$
- Q1(vi): Check whether the following are quadratic equations : (vi) $x^2 + 3x + 1 = (x – 2)^2$
- Q1(vii): Check whether the following are quadratic equations : (vii) $(x + 2)^3 = 2x (x^2 – 1)$
- Q1(viii): Check whether the following are quadratic equations : (viii) $x^3 – 4x^2 – x + 1 = (x – 2)^3$
- Q2(i): Represent the following situations in the form of quadratic equations : (i) The area of a rectangular plot is $528$ $m^2$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
- Q2(iii): Represent the following situations in the form of quadratic equations : (iii) Rohan’s mother is $26$ years older than him. The product of their ages (in years) $3$ years from now will be $360$. We would like to find Rohan’s present age.
- Q2(iv): Represent the following situations in the form of quadratic equations : (iv) A train travels a distance of $480$ km at a uniform speed. If the speed had been $8$ km/h less, then it would have taken $3$ hours more to cover the same distance. We need to find the speed of the train.
