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Q2(ii):
Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots. (ii) $kx (x – 2) + 6 = 0$
Solution :
Given: A quadratic equation $kx(x - 2) + 6 = 0$, where $k \neq 0$.
To Find: The value(s) of $k$ such that the quadratic equation has two equal real roots.
Step 1: Simplifying the equation into standard form
The standard form of a quadratic equation is $ax^2 + bx + c = 0$. We expand the given equation:
$kx(x - 2) + 6 = 0$
$kx^2 - 2kx + 6 = 0$
[Distributive property of multiplication over subtraction]
Step 2: Identifying coefficients
Comparing $kx^2 - 2kx + 6 = 0$ with $ax^2 + bx + c = 0$, we identify:
$a = k$
$b = -2k$
$c = 6$
Step 3: Applying the condition for equal roots
For a quadratic equation to have two equal real roots, its discriminant ($D$) must be equal to zero.
Formula: $D = b^2 - 4ac = 0$
Substituting the identified coefficients into the discriminant formula:
$(-2k)^2 - 4(k)(6) = 0$
Step 4: Solving for $k$
$4k^2 - 24k = 0$
[Simplifying the powers and products]
Factor out the common term $4k$:
$4k(k - 6) = 0$
[Using the zero-product property: if $p \cdot q = 0$, then $p=0$ or $q=0$]
Case 1: $4k = 0 \implies k = 0$
Case 2: $k - 6 = 0 \implies k = 6$
Step 5: Verification of validity
If $k = 0$, the equation becomes $0x^2 - 2(0)x + 6 = 0$, which simplifies to $6 = 0$. This is a contradiction and not a quadratic equation. Therefore, $k$ cannot be $0$.
If $k = 6$, the equation becomes $6x^2 - 12x + 6 = 0$. Dividing by 6, we get $x^2 - 2x + 1 = 0$, which is $(x - 1)^2 = 0$, yielding equal roots $x = 1, 1$.
Final Answer: The value of $k$ is 6.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.3
- Q1(i): Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (i) $2x^2 – 3x + 5 = 0$
- Q1(ii): Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (ii) $3x^2 – 4\sqrt{3}x + 4 = 0$
- Q1(iii): Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (iii) $2x^2 – 6x + 3 = 0$
- Q2(i): Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots. (i) $2x^2 + kx + 3 = 0$
- Q3: Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is $800$ $m^2$? If so, find its length and breadth.
- Q4: Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is $20$ years. Four years ago, the product of their ages in years was $48$.
- Q5: Is it possible to design a rectangular park of perimeter $80$ m and area $400$ $m^2$? If so, find its length and breadth.
CBSE Solutions for Class 10 Mathematics Quadratic Equations
Chapters in CBSE - Class 10 Mathematics
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