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Q1(vi):
Check whether the following are quadratic equations : (vi) $x^2 + 3x + 1 = (x – 2)^2$
Solution :
Given: The equation $x^2 + 3x + 1 = (x - 2)^2$.
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: Expand the right-hand side of the equation.
The given equation is:
$x^2 + 3x + 1 = (x - 2)^2$
We use the algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = x$ and $b = 2$.
$(x - 2)^2 = x^2 - 2(x)(2) + (2)^2$
$(x - 2)^2 = x^2 - 4x + 4$
Step 2: Substitute the expanded form back into the original equation.
$x^2 + 3x + 1 = x^2 - 4x + 4$
Step 3: Rearrange the terms to one side to set the equation to zero.
Subtract $(x^2 - 4x + 4)$ from both sides of the equation:
$x^2 + 3x + 1 - (x^2 - 4x + 4) = 0$
$x^2 + 3x + 1 - x^2 + 4x - 4 = 0$
Step 4: Simplify the expression by combining like terms.
Group the $x^2$ terms, the $x$ terms, and the constant terms:
$(x^2 - x^2) + (3x + 4x) + (1 - 4) = 0$
$0x^2 + 7x - 3 = 0$
$7x - 3 = 0$
Step 5: Analyze the resulting equation.
The simplified equation is $7x - 3 = 0$. This is a linear equation because the highest power of the variable $x$ is 1. Comparing this to the standard form $ax^2 + bx + c = 0$, we see that the coefficient of $x^2$ (which is $a$) is $0$. Since a quadratic equation must have $a \neq 0$, this equation does not satisfy the definition.
Final Answer: The given equation $x^2 + 3x + 1 = (x - 2)^2$ is not a quadratic equation because the $x^2$ terms cancel out, leaving a linear equation.
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(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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