Question

The value of expression 1+a+a²

Answer 1

Answer:

1+a+a²

= 1+a=2a +a²

answer=2a+a²

Answer 2

The value of the expression a²+a+1  is $\alpha = \frac{-1+i\sqrt{3 } }{2}$ and  $\beta = \frac{-1-i\sqrt{3 } }{2}$

Explanation:

Given:

a²+a+1

To find:

The roots

Formula:

$\alpha = \frac{-b+\sqrt{b^{2}-4ac } }{2a}$

$\beta= \frac{-b-\sqrt{b^{2}-4ac } }{2a}$

==> a²+a+1

==> a= Coefficient of a² = 1

==> b=Coefficient of a = 1

==>c= Constant = 1

Substitute the values in the formula:

$\alpha = \frac{-b+\sqrt{b^{2}-4ac } }{2a}$

$\alpha = \frac{-1+\sqrt{1^{2}-4(1)(1) } }{2(1)}$

$\alpha = \frac{-1+\sqrt{1-4 } }{2}$

$\alpha = \frac{-1+\sqrt{-3 } }{2}$

We know that, i²=-1

$\alpha = \frac{-1+\sqrt{-1\times3 } }{2}$

$\alpha = \frac{-1+\sqrt{i^{2} \times3 } }{2}$

$\alpha = \frac{-1+i\sqrt{3 } }{2}$

Substitute the values in the formula:

$\beta = \frac{-b-\sqrt{b^{2}-4ac } }{2a}$

$\beta = \frac{-1-\sqrt{1^{2}-4(1)(1) } }{2(1)}$

$\beta = \frac{-1-\sqrt{1-4 } }{2}$

$\beta = \frac{-1-\sqrt{-3 } }{2}$

We know that, i²=-1

$\beta = \frac{-1-\sqrt{-1\times3 } }{2}$

$\beta = \frac{-1-\sqrt{i^{2} \times3 } }{2}$

$\beta = \frac{-1-i\sqrt{3 } }{2}$

The value of the expression a²+a+1  is $\alpha = \frac{-1+i\sqrt{3 } }{2}$ and  $\beta = \frac{-1-i\sqrt{3 } }{2}$