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Q1(ii):
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (ii) $3x^2 – 4\sqrt{3}x + 4 = 0$
Solution :
Given: A quadratic equation $3x^2 - 4\sqrt{3}x + 4 = 0$.
To Find: The nature of the roots and the roots themselves if they exist.
Step 1: Identify the coefficients of the quadratic equation.
The standard form of a quadratic equation is $ax^2 + bx + c = 0$. Comparing the given equation $3x^2 - 4\sqrt{3}x + 4 = 0$ with the standard form, we identify:
$a = 3$
$b = -4\sqrt{3}$
$c = 4$
Step 2: Determine the nature of the roots using the Discriminant ($D$).
The discriminant is given by the formula $D = b^2 - 4ac$.
Substituting the values identified in Step 1:
$D = (-4\sqrt{3})^2 - 4(3)(4)$
$D = (16 \times 3) - 48$
$D = 48 - 48$
$D = 0$
[Since $D = 0$, the quadratic equation has two equal real roots.]
Step 3: Calculate the roots using the Quadratic Formula.
The quadratic formula is given by $x = \frac{-b \pm \sqrt{D}}{2a}$.
Since $D = 0$, the formula simplifies to $x = \frac{-b}{2a}$.
Substituting the values:
$x = \frac{-(-4\sqrt{3})}{2(3)}$
$x = \frac{4\sqrt{3}}{6}$
Step 4: Simplify the expression.
$x = \frac{4\sqrt{3}}{6}$
Dividing both the numerator and the denominator by their greatest common divisor, which is $2$:
$x = \frac{2\sqrt{3}}{3}$
Since the roots are equal, both roots are $\frac{2\sqrt{3}}{3}$.
[Note: $\frac{2\sqrt{3}}{3}$ can also be written as $\frac{2}{\sqrt{3}}$ by rationalizing the denominator.]
Final Answer: The roots are real and equal, and the roots are $\frac{2\sqrt{3}}{3}, \frac{2\sqrt{3}}{3}$.
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(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) $2x^2 + kx + 3 = 0$
- 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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