Question

What are all the values of x which satisfy ...? (Check photo for correct numbering)

Answer 1

x = -6, 0, and 1

Explanation:

We are tasked with finding the values of 'x' that satisfy the given equation. To solve this, we will analyze the properties of exponential functions and equate the exponent to zero, then solve for 'x'.

Given equation:

`4^(x^3+5x^2-6x)=1`

`\hrulefill`

Solving the equation for 'x'.

Take the natural logarithm of each side of the equation:

`\Longrightarrow 4^(x^3+5x^2-6x)=1\\\\\\\\\Longrightarrow \ln(4^(x^3+5x^2-6x))=\ln(1)\\\\\\\\\Longrightarrow (x^3+5x^2-6x)\cdot\ln(4)=0`

Factor the polynomial:

`\Longrightarrow \ln(4)(x^3+5x^2-6x)=0\\\\\\\\\Longrightarrow \ln(4)[x(x-1)(x+6)]=0\\\\\\\\\Longrightarrow x(x-1)(x+6)=0\\\\\\\\\therefore \boxed{x=-6, \ 0, \ 1}`

Thus, we get three values of 'x' that satisfy the equation.