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Q1(i):
Check whether the following are quadratic equations : (i) $(x + 1)^2 = 2(x – 3)$
Solution :
Given: The equation $(x + 1)^2 = 2(x - 3)$.
To Find: Determine whether the given equation is a quadratic equation.
Definition: A quadratic equation in the variable $x$ is an equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$.
Step 1: Expanding the left-hand side (LHS) of the equation.
The expression is $(x + 1)^2$. We use the algebraic identity $(a + b)^2 = a^2 + 2ab + b^2$.
Substituting $a = x$ and $b = 1$:
$(x + 1)^2 = x^2 + 2(x)(1) + (1)^2$
$(x + 1)^2 = x^2 + 2x + 1$
Step 2: Expanding the right-hand side (RHS) of the equation.
The expression is $2(x - 3)$. We use the distributive property $a(b - c) = ab - ac$.
$2(x - 3) = 2(x) - 2(3)$
$2(x - 3) = 2x - 6$
Step 3: Equating the expanded sides and simplifying.
Equating the results from Step 1 and Step 2:
$x^2 + 2x + 1 = 2x - 6$
Step 4: Bringing all terms to one side to form the standard quadratic form.
Subtract $2x$ from both sides:
$x^2 + 2x - 2x + 1 = -6$
$x^2 + 1 = -6$
Add $6$ to both sides:
$x^2 + 1 + 6 = 0$
$x^2 + 7 = 0$
Step 5: Comparing with the standard form $ax^2 + bx + c = 0$.
The equation $x^2 + 7 = 0$ can be written as $1x^2 + 0x + 7 = 0$.
Here, $a = 1$, $b = 0$, and $c = 7$.
Since $a \neq 0$ (as $1 \neq 0$), the equation satisfies the condition for being a quadratic equation.
Final Answer: Yes, the given equation $(x + 1)^2 = 2(x - 3)$ is a quadratic equation because it can be simplified to the form $ax^2 + bx + c = 0$ where $a \neq 0$.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.1
- 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(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(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.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
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