Question

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

Answer 1

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

$i=\sqrt{-1}$

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

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

Answer 2

Answer:

B,C,E,F

Step-by-step explanation: