Question

Identify all of the root(s) of g(x) = (x2 + 3x - 4)(x2 - 4x + 29).

Answer 1

B,C,E,F

Explanation:

Answer 2

we have

`g(x)=(x^(2)+3x-4)( x^(2)-4x+29)`

To find the roots of g(x)

Find the roots of the first term and then find the roots of the second term

Step 1

Find the roots of the first term

`(x^(2)+3x-4)=0`

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`(x^(2)+3x)=4`

Complete the square. Remember to balance the equation by adding the same constants to each side

`(x^(2)+3x+1.5^(2))=4+1.5^(2)`

`(x^(2)+3x+1.5^(2))=6.25`

Rewrite as perfect squares

`(x+1.5)^(2)=6.25`

Square root both sides

`(x+1.5)=(+/-)2.5`

`x=-1.5(+/-)2.5`

`x=-1.5+2.5=1`

`x=-1.5-2.5=-4`

so the factored form of the first term is

`(x^(2)+3x-4)=(x-1)(x+4)`

Step 2

Find the roots of the second term

`(x^(2)-4x+29)=0`

Group terms that contain the same variable, and move the constant to the opposite side of the equation

`(x^(2)-4x)=-29`

Complete the square. Remember to balance the equation by adding the same constants to each side

`(x^(2)-4x+4)=-29+4`

`(x^(2)-4x+4)=-25`

Rewrite as perfect squares

`(x-2)^(2)=-25`

Remember that

`i=√(-1)`

Square root both sides

`(x-2)=(+/-)5i`

`x=2(+/-)5i`

`x=2+5i`

`x=2-5i`

so the factored form of the second term is

`(x^(2)-4x+29)=(x-(2+5i))(x-(2-5i))`

Step 3

Substitute the factored form of the first and second term in g(x)

`g(x)=(x-1)(x+4)(x-(2+5i))(x-(2-5i))`

therefore

the answer is

the roots are

`x1=1\\x2=-4\\x3=(2+5i)\\x4=(2-5i)`