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Q1(i):
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (i) $2x^2 – 3x + 5 = 0$
Solution :
Given: A quadratic equation $2x^2 - 3x + 5 = 0$.
To Find: The nature of the roots of the given quadratic equation and, if real roots exist, to determine 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 $2x^2 - 3x + 5 = 0$ with the standard form:
$a = 2$
$b = -3$
$c = 5$
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 $c$ into the formula:
$D = (-3)^2 - 4(2)(5)$
$D = 9 - 40$
$D = -31$
Step 3: Analyze the nature of the roots based on the value of $D$.
According to the theory of quadratic equations:
1. If $D > 0$, the equation has two distinct real roots.
2. If $D = 0$, the equation has two equal real roots.
3. If $D < 0$, the equation has no real roots (the roots are complex/imaginary).
Since $D = -31$, and $-31 < 0$, the discriminant is negative.
Step 4: Conclusion regarding the existence of real roots.
Because the discriminant is less than zero ($D < 0$), the square root of the discriminant would result in an imaginary number. Therefore, the quadratic equation $2x^2 - 3x + 5 = 0$ does not possess any real roots.
Final Answer: The discriminant is $-31$, which is less than $0$. Therefore, the given quadratic equation has no real roots.
More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.3
- 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) $2x^2 + kx + 3 = 0$
- 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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