Question

Find the quadratic polynomial each with the given number as the sum and product of its zero :-1/4,1/4​

Answer 1

Step-by-step explanation:

Given -

Sum of zeroes = 1/4

Product of zeroes = 1/4

To Find -

• A quadratic polynomial

As we know that :-

• α + β = -b/a

→ 1/4 = -b/a ..... (i)

And

• αβ = c/a

→ 1/4 = c/a ...... (ii)

Now, From (i) and (ii), we get :

a = 4

b = -1

c = 1

As we know that :-

For a quadratic polynomial :-

• ax² + bx + c

→ (4)x² + (-1)x + (1)

→ 4x² - x + 1

Hence,

The polynomial is 4x² - x + 1

Verification :-

• α + β = -b/a

→ 1/4 = -(-1)/4

→ 1/4 = 1/4

LHS = RHS

And

• αβ = c/a

→ 1/4 = 1/4

LHS = RHS

Hence,

Verified...

It shows that our answer is absolutely correct.

Answer 2

$\large{\underline{\bf{\purple{Given:-}}}}$

• ✦ sum of zeroes = 1/4

• ✦ product of zeroes = 1/4

$\large{\underline{\bf{\purple{To\:Find:-}}}}$

✦ we need to find the quadratic polynomial.

$\huge{\underline{\bf{\red{Solution:-}}}}$

Let α and β are the zeroes of quadratic polynomial.

• ( α + β ) = 1/4 [ Given]

• αβ = 1/4 [Given]

Then

$\leadsto \rm\:\:$ x²- (α + β)x + αβ

$\leadsto \rm\:\:x^2-\frac{1}{4}x \: + \: \frac{1}{4} = 0 \\ \\\leadsto \rm\:\: \frac{ {4x}^{2} - 1 x+ 1}{4} = 0 \\ \\ \leadsto \rm\:\:4 {x}^{2} -x + 1 = 4 \times 0 \\ \\\leadsto \bf{ \pink{\:\:4 {x}^{2} - x + 1 = 0}}$

Hence the quadratic polynomial is

⠀⠀⠀⠀⠀ 4x² - x +1 =0.

Now,

Verification :-

• a = 4

• b = -1

• c = 1

sum of zeroes = - b/a

(α + β) = - b/a

$\leadsto \rm\:\frac{1}{4}=\frac{-(1)}{4}$

$\leadsto \rm\:\:\frac{1}{4}= \frac{1}{4}$

Product of zeroes = c/a

αβ = c/a

$\leadsto \rm\:\:\frac{1}{4}=\frac{1}{4}$

LHS = RHS

Hence Verified.

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