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Q1(iv):
Check whether the following are quadratic equations : (iv) $(x – 3)(2x +1) = x(x + 5)$
Solution :
Given: The algebraic equation $(x - 3)(2x + 1) = x(x + 5)$.
To Find: Determine whether the given equation is a quadratic equation.
Definition: A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$. The highest power (degree) of the variable in a quadratic equation must be 2.
Step 1: Expanding the Left-Hand Side (LHS)
The LHS is $(x - 3)(2x + 1)$. We apply the distributive property (FOIL method):
$(x - 3)(2x + 1) = x(2x) + x(1) - 3(2x) - 3(1)$
$= 2x^2 + x - 6x - 3$
$= 2x^2 - 5x - 3$ [Combining like terms $x$ and $-6x$]
Step 2: Expanding the Right-Hand Side (RHS)
The RHS is $x(x + 5)$. We apply the distributive property:
$x(x + 5) = x(x) + x(5)$
$= x^2 + 5x$
Step 3: Equating LHS and RHS and Simplifying
Now, set the expanded LHS equal to the expanded RHS:
$2x^2 - 5x - 3 = x^2 + 5x$
To bring the equation into the standard form $ax^2 + bx + c = 0$, we subtract $(x^2 + 5x)$ from both sides:
$(2x^2 - x^2) + (-5x - 5x) - 3 = 0$
$x^2 - 10x - 3 = 0$
Step 4: Verification against the Standard Form
Comparing $x^2 - 10x - 3 = 0$ with the standard form $ax^2 + bx + c = 0$:
Here, $a = 1$, $b = -10$, and $c = -3$.
Since $a \neq 0$ and the highest degree of the variable $x$ is 2, the equation satisfies the definition of a quadratic equation.
Final Answer: Yes, the given equation $(x - 3)(2x + 1) = x(x + 5)$ is a quadratic equation because it simplifies to the form $x^2 - 10x - 3 = 0$.
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(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.
