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Q2(i):
Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots. (i) $2x^2 + kx + 3 = 0$
Solution :
Given: A quadratic equation $2x^2 + kx + 3 = 0$.
To find: The value(s) of $k$ such that the given quadratic equation has two equal real roots.
Step 1: Identify the coefficients of the quadratic equation.
A standard quadratic equation is represented as $ax^2 + bx + c = 0$. Comparing the given equation $2x^2 + kx + 3 = 0$ with the standard form, we identify:
$a = 2$
$b = k$
$c = 3$
Step 2: State the condition for equal roots.
For a quadratic equation $ax^2 + bx + c = 0$, the nature of the roots is determined by the discriminant, denoted by $D$, where $D = b^2 - 4ac$.
The condition for a quadratic equation to have two equal real roots is that the discriminant must be equal to zero:
$D = b^2 - 4ac = 0$
Step 3: Substitute the coefficients into the discriminant formula.
Substituting $a = 2$, $b = k$, and $c = 3$ into the equation $b^2 - 4ac = 0$:
$(k)^2 - 4(2)(3) = 0$
Step 4: Solve the resulting equation for $k$.
Perform the multiplication inside the expression:
$k^2 - 24 = 0$
Isolate $k^2$ by adding 24 to both sides of the equation:
$k^2 = 24$
Take the square root of both sides to solve for $k$:
$k = \pm\sqrt{24}$
Simplify the radical expression $\sqrt{24}$:
$\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$
Therefore:
$k = \pm 2\sqrt{6}$
Step 5: Conclusion.
The values of $k$ for which the quadratic equation $2x^2 + kx + 3 = 0$ has two equal roots are $2\sqrt{6}$ and $-2\sqrt{6}$.
Final Answer: k = 2\sqrt{6}, -2\sqrt{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(ii): Find the values of $k$ for each of the following quadratic equations, so that they have two equal roots. (ii) $kx (x – 2) + 6 = 0$
- 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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