Question

find the zeroes of the polynomial and verify the relationship between zeroes and coifficient​ x^2- root 2 x- 3/2​

Answer 1

Solution :--

——»› Quadratic polynomial ‹«——

${x}^{2} - \sqrt{2} x - \frac{3}{2} = 0$

• ————-—————————

${x}^{2} - \sqrt{2} x - \frac{3}{2} = 0 \\ multiplying \: by \: 2 \: on \: both \: the \: side \\ 2 {x}^{2} - 2 \sqrt{2} x - 3 = 0 \\ by \: facto risation \: method \\ \\ 2 {x}^{2} - 3 \sqrt{2} x + \sqrt{2} x - 3 = 0 \\ (2 {x}^{2} - 3 \sqrt{2} x)( \sqrt{2} x - 3) = 0 \\ \sqrt{2}x( \sqrt{2} x - 3) 1( \sqrt{2} x - 3) = 0 \\ ( \sqrt{2}x - 3)( \sqrt{2} x + 1) = 0 \\ ( \sqrt{2}x - 3) = 0 \: \: \: or \: \: \: ( \sqrt{2} x + 1) = 0 \\ \sqrt{2} x = 3 \: \: \: or \: \: \: \sqrt{2} x = - 1 \\ x = \frac{3}{ \sqrt{2} } \: \: \: \: \: or \: \: \: \: \: x = \frac{ - 1}{ \sqrt{2} }$

If the coefficient of x^2 is not equal to the 1 then the change the sign of found factors and divide by Coefficient of x^2

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Let's see

We find the factors are. ,

$- 3 \sqrt{2} , \sqrt{2} \\ change \: the \: sign \\3 \sqrt{2}, - \sqrt{2} \\ now \: devide \: by \: coefficient \: of \: {x}^{2} \\ i.e. \: \: \: 2 \\ \frac{3 \sqrt{2} }{2} , \: \frac{ - \sqrt{2} }{2} \\ \frac{3}{ \sqrt{2} } , \frac{ - 1}{ \sqrt{2} }$

This is a relationship between zeroes and coifficient

———»› thanks for the question bro ❤️✍️✌️

Answer 2

Step-by-step explanation:

Given -

• p(x) = x² - √2x - 3/2

To Find -

• Zeroes of the polynomial

Now,

→ x² - √2x - 3/2 = 0

→ 2x² - 2√2x - 3/2 = 0

→ 2x² - 2√2x - 3 = 0

Now, Factorising this, we get :

By Quadratic formula :

• x = -b ± √b² - 4ac/2a

→ -(-2√2) ± √(-2√2)² - 4×2×-3/2(2)

→ 2√2 ± √8 + 24/4

→ 2√2 ± √32/4

→ 2√2 ± 4√2/4

Zeroes are -

→ x = 2√2 + 4√2/4

→ 6√2/4

• → 3√2/2

And

→ x = 2√2 - 4√2/4

→ -2√2/4

• → -√2/2

Verification :-

As we know that :-

• α + β = -b/a

→ 3√2/2 + (-√2/2) = -(-2√2)/2

→ 3√2/2 - √2/2 = 2√2/2

→ 3√2 - √2/2 = 2√2/2

→ 2√2/2 = √2

→ √2 = √2

LHS = RHS

And

• αβ = c/a

→ 3√2/2 × -√2/2 = -3/2

→ -6/4 = -3/2

→ -3/2 = -3/2

LHS = RHS

Hence,

Verified..

Formula Used :-

☞ Quadratic formula :-

• x = -b ± √b² - 4ac/2a