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Q1(iii):
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (iii) $2x^2 – 6x + 3 = 0$
Solution :
Given: A quadratic equation $2x^2 - 6x + 3 = 0$.
To Find: The nature of the roots of the given quadratic equation and, if real roots exist, find their values.
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 $2x^2 - 6x + 3 = 0$ with the standard form:
- $a = 2$
- $b = -6$
- $c = 3$
Step 2: Determine the nature of the roots using the Discriminant ($D$).
The discriminant of a quadratic equation is given by the formula $D = b^2 - 4ac$.
Substituting the values of $a$, $b$, and $c$:
$D = (-6)^2 - 4(2)(3)$
$D = 36 - 24$
$D = 12$
[Since $D > 0$, the quadratic equation has two distinct real roots.]
Step 3: Apply the Quadratic Formula to find the roots.
The quadratic formula is given by $x = \frac{-b \pm \sqrt{D}}{2a}$.
Substituting the known values into the formula:
$x = \frac{-(-6) \pm \sqrt{12}}{2(2)}$
$x = \frac{6 \pm \sqrt{4 \times 3}}{4}$
$x = \frac{6 \pm 2\sqrt{3}}{4}$
Step 4: Simplify the expression.
Factor out the common term $2$ from the numerator:
$x = \frac{2(3 \pm \sqrt{3})}{4}$
$x = \frac{3 \pm \sqrt{3}}{2}$
Therefore, the two roots are:
$x_1 = \frac{3 + \sqrt{3}}{2}$
$x_2 = \frac{3 - \sqrt{3}}{2}$
Final Answer: The roots are real and distinct. The roots of the equation are $\frac{3 + \sqrt{3}}{2}$ and $\frac{3 - \sqrt{3}}{2}$.
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) $3x^2 – 4\sqrt{3}x + 4 = 0$
- 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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