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Q2(iii):
Represent the following situations in the form of quadratic equations : (iii) Rohanβs 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 Rohanβs present age.
Solution :
Given:
1. Rohanβs mother is $26$ years older than Rohan.
2. The product of their ages $3$ years from now will be $360$.
To Find:
Represent the given situation in the form of a quadratic equation in terms of Rohan's present age.
Step 1: Defining the Variables
Let the present age of Rohan be $x$ years.
Since Rohanβs mother is $26$ years older than him, the present age of Rohanβs mother is $(x + 26)$ years.
Step 2: Determining Ages 3 Years from Now
After $3$ years, the age of each person will increase by $3$ years.
Rohanβs age after $3$ years = $(x + 3)$ years.
Rohanβs motherβs age after $3$ years = $(x + 26) + 3 = (x + 29)$ years.
Step 3: Formulating the Equation
According to the problem, the product of their ages $3$ years from now is $360$.
Therefore, we set up the equation:
$(x + 3)(x + 29) = 360$
Step 4: Expanding and Simplifying the Equation
Using the distributive property (FOIL method) to expand the left side:
$x(x + 29) + 3(x + 29) = 360$
$x^2 + 29x + 3x + 87 = 360$
[Combining like terms $29x$ and $3x$]:
$x^2 + 32x + 87 = 360$
Step 5: Bringing the Equation to Standard Form
The standard form of a quadratic equation is $ax^2 + bx + c = 0$. Subtract $360$ from both sides:
$x^2 + 32x + 87 - 360 = 0$
$x^2 + 32x - 273 = 0$
Justification:
The resulting equation $x^2 + 32x - 273 = 0$ is a polynomial of degree $2$, which satisfies the definition of a quadratic equation where $a=1$, $b=32$, and $c=-273$.
Final Answer: The quadratic equation representing the situation is $x^2 + 32x - 273 = 0$, where $x$ is Rohan's present age.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.1
- Q1(i): Check whether the following are quadratic equations : (i) $(x + 1)^2 = 2(x β 3)$
- Q1(ii): Check whether the following are quadratic equations : (ii) $x^2 β 2x = (β2)(3 β x)$
- Q1(iii): Check whether the following are quadratic equations : (iii) $(x β 2)(x + 1) = (x β 1)(x + 3)$
- Q1(iv): Check whether the following are quadratic equations : (iv) $(x β 3)(2x +1) = x(x + 5)$
- Q1(v): Check whether the following are quadratic equations : (v) $(2x β 1)(x β 3) = (x + 5)(x β 1)$
- Q1(vi): Check whether the following are quadratic equations : (vi) $x^2 + 3x + 1 = (x β 2)^2$
- Q1(vii): Check whether the following are quadratic equations : (vii) $(x + 2)^3 = 2x (x^2 β 1)$
- Q1(viii): Check whether the following are quadratic equations : (viii) $x^3 β 4x^2 β x + 1 = (x β 2)^3$
- 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(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.
- 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.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
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