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Q1(ii):
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(ii) $4s^2 - 4s + 1$
Solution :
Given: A quadratic polynomial $p(s) = 4s^2 - 4s + 1$.
To Find:
1. The zeroes of the polynomial $p(s)$.
2. Verification of the relationship between the zeroes and the coefficients of the polynomial.
Step 1: Finding the zeroes of the polynomial
To find the zeroes of $p(s)$, we set $p(s) = 0$.
$4s^2 - 4s + 1 = 0$
We use the splitting the middle term method. We look for two numbers whose product is $4 \times 1 = 4$ and whose sum is $-4$. These numbers are $-2$ and $-2$.
$4s^2 - 2s - 2s + 1 = 0$
Now, factor by grouping:
$2s(2s - 1) - 1(2s - 1) = 0$
$(2s - 1)(2s - 1) = 0$
Setting each factor to zero:
$2s - 1 = 0 \implies s = \frac{1}{2}$
$2s - 1 = 0 \implies s = \frac{1}{2}$
Thus, the zeroes of the polynomial are $\alpha = \frac{1}{2}$ and $\beta = \frac{1}{2}$.
Step 2: Identifying coefficients
Comparing $4s^2 - 4s + 1$ with the standard form $as^2 + bs + c$:
$a = 4$
$b = -4$
$c = 1$
Step 3: Verification of the relationship between zeroes and coefficients
The relationship states:
1. Sum of zeroes ($\alpha + \beta$) = $-\frac{b}{a}$
2. Product of zeroes ($\alpha \cdot \beta$) = $\frac{c}{a}$
Verification of Sum of Zeroes:
$\alpha + \beta = \frac{1}{2} + \frac{1}{2} = 1$
$-\frac{b}{a} = -\frac{(-4)}{4} = \frac{4}{4} = 1$
Since $1 = 1$, the sum of zeroes is verified.
Verification of Product of Zeroes:
$\alpha \cdot \beta = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$
$\frac{c}{a} = \frac{1}{4}$
Since $\frac{1}{4} = \frac{1}{4}$, the product of zeroes is verified.
Final Answer: The zeroes of the polynomial $4s^2 - 4s + 1$ are $\frac{1}{2}$ and $\frac{1}{2}$. The relationship between the zeroes and coefficients is verified as the sum of zeroes is $1$ and the product of zeroes is $\frac{1}{4}$.
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(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) $t^2 - 15$
- Q1(vi): Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. (vi) $3x^2 - x - 4$
- 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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