Question

if alpha and beeta are the zero of the polynomial p(x)=3x2--12x++15 find the value of alpha square ++beeta square​

Answer 1

Solution :

$\bf{\green{\underline{\underline{\bf{Given\::}}}}}$

If α and β are the zeroes of the polynomial p(x) = 3x² - 12x + 15.

$\bf{\green{\underline{\underline{\bf{To\:find\::}}}}}$

The value of α² + β².

$\bf{\green{\underline{\underline{\bf{Explanation\::}}}}}$

We have quadratic polynomial as we compared with ax² + bx + c = 0;

• a = 3
• b = -12
• c = 15

$\bf{\orange{\underline{\underline{\mathcal{SUM\:OF\:THE\:ZEROES\::}}}}}$

$\mapsto\sf{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:(x)^{2} }{Coefficient\:of\:x} }\\\\\\\mapsto\sf{\alpha +\beta =\dfrac{-(-12)}{3} }\\\\\\\mapsto\sf{\alpha +\beta =\cancel{\dfrac{12}{3}} }\\\\\\\mapsto\sf{\red{\alpha +\beta =4}}$

$\bf{\orange{\underline{\underline{\mathcal{PRODUCT\:OF\:THE\:ZEROES\::}}}}}$

$\mapsto\sf{\alpha \times \beta =\dfrac{c}{a} =\dfrac{Constant\:term }{Coefficient\:of\:x} }\\\\\\\mapsto\sf{\alpha \times \beta =\dfrac{15}{3} }\\\\\\\mapsto\sf{\alpha \times \beta =\cancel{\dfrac{15}{3}} }\\\\\\\mapsto\sf{\red{\alpha \times \beta =5}}$

Now;

$\mapsto\sf{\alpha^{2} +\beta ^{2} =(\alpha +\beta)^{2} -2\alpha \beta }\\\\\mapsto\sf{\alpha^{2} +\beta^{2} =(4 )^{2} -2(5) }\\\\\mapsto\sf{\alpha^{2} +\beta^{2} =16 -10}\\\\\mapsto\alpha^{2} +\beta ^{2}=16-10}\\ \\\mapsto\sf{\orange{\alpha^{2}+ \beta ^{2} =6}}$

Answer 2

Step-by-step explanation:

Given -

α and β are zeroes of the polynomial 3x² - 12x + 15

To Find -

Value of α² + β²

Now,

3x² - 12x + 15

here,

a = 3

b = - 12

c = 15

Then,

α + β = - b/a

α + β = -(-12)/3

α + β = 12/3

• α + β = 4

Now,

Squaring both sides

(α + β)² = (4)²

= α² + 2αβ + β² = 16

Adding - 2αβ both sides

= α² + 2αβ + β² - 2αβ = 16 - 2αβ

• = α² + β² = 16 - 2αβ

Now,

αβ = c/a

αβ = 15/3

• = αβ = 5

Multiplying 2 both sides

2αβ = 5 × 2

• = 2αβ = 10

Now,

Substituting the value of 2αβ on

α² + β² = 16 - 2αβ

= α² + β² = 16 - 10

= α² + β² = 6

Hence,

The value of α² + β² is 6.