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Q1(vi):
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(vi) $3x^2 - x - 4$
Solution :
Given: A quadratic polynomial $p(x) = 3x^2 - x - 4$.
To Find: The zeroes of the polynomial and verify the relationship between the zeroes and the coefficients of the polynomial.
Step 1: Finding the zeroes of the polynomial
To find the zeroes, we set $p(x) = 0$.
$3x^2 - x - 4 = 0$
We use the splitting the middle term method. We need two numbers whose product is $3 \times (-4) = -12$ and whose sum is $-1$. These numbers are $-4$ and $3$.
$3x^2 - 4x + 3x - 4 = 0$ [Splitting the middle term $-x$ into $-4x + 3x$]
$x(3x - 4) + 1(3x - 4) = 0$ [Factoring by grouping]
$(3x - 4)(x + 1) = 0$ [Taking $(3x - 4)$ as a common factor]
Setting each factor to zero:
1) $3x - 4 = 0 \implies 3x = 4 \implies x = \frac{4}{3}$
2) $x + 1 = 0 \implies x = -1$
Thus, the zeroes are $\alpha = \frac{4}{3}$ and $\beta = -1$.
Step 2: Identifying coefficients
Comparing $3x^2 - x - 4$ with the standard form $ax^2 + bx + c$:
$a = 3$
$b = -1$
$c = -4$
Step 3: Verifying the relationship between zeroes and coefficients
The relationships to verify are:
1) Sum of zeroes ($\alpha + \beta$) = $-\frac{b}{a}$
2) Product of zeroes ($\alpha \cdot \beta$) = $\frac{c}{a}$
Verification of Sum of Zeroes:
LHS: $\alpha + \beta = \frac{4}{3} + (-1) = \frac{4}{3} - \frac{3}{3} = \frac{1}{3}$
RHS: $-\frac{b}{a} = -\frac{(-1)}{3} = \frac{1}{3}$
Since LHS = RHS, the relationship is verified.
Verification of Product of Zeroes:
LHS: $\alpha \cdot \beta = \left(\frac{4}{3}\right) \cdot (-1) = -\frac{4}{3}$
RHS: $\frac{c}{a} = \frac{-4}{3} = -\frac{4}{3}$
Since LHS = RHS, the relationship is verified.
Final Answer: The zeroes of the polynomial $3x^2 - x - 4$ are $\frac{4}{3}$ and $-1$. The relationship between the zeroes and coefficients is verified as the sum of zeroes is $\frac{1}{3}$ and the product of zeroes is $-\frac{4}{3}$.
More Questions from Class 10 Mathematics Polynomials EXERCISE 2.2
- Q1(i): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (i) $x^2 - 2x - 8$
- Q1(ii): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (ii) $4s^2 - 4s + 1$
- Q1(iii): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (iii) $6x^2 - 3 - 7x$
- Q1(iv): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (iv) $4u^2 + 8u$
- Q1(v): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (v) $t^2 - 15$
- Q2(i): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (i) $\frac{1}{4}, -1$
- Q2(ii): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (ii) $\sqrt{2}, \frac{1}{3}$
- Q2(iii): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (iii) $0, \sqrt{5}$
- Q2(iv): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (iv) $1, 1$
- Q2(v): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (v) $-\frac{1}{4}, \frac{1}{4}$
- Q2(vi): Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively. (vi) $4, 1$
CBSE Solutions for Class 10 Mathematics Polynomials
Chapters in CBSE - Class 10 Mathematics
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