Question

What is are the solutions to the statement:(2x-7)(4x^2+14x+49)=0

Answer 1

Given the question, the following are the solution steps to answer the question.

STEP 1: Write the given equation.

`(2x-7)(4x^2+14x+49)=0`

STEP 2: Find the solutions to the statement

First splitting the equation into two, we have:

`\begin{gathered} (2x-7)(4x^(2)+14x+49)=0 \\ 2x-7=0 \\ 4x^2+14x+49=0 \\ \\ 2x-7=0 \\ 2x=7 \\ x=(7)/(2) \end{gathered}`

STEP 3: Find the solution to the second equation

`\begin{gathered} 4x^2+14x+49=0 \\ Using\text{ quadratic formula:} \\ x_(1,\:2)=(-b\pm√(b^2-4ac))/(2a) \\ From\text{ the equation,} \\ a=4,b=14,c=49 \\ \\ By\text{ substitution,} \\ x_(1,\:2)=(-14\pm √(14^2-4\cdot \:4\cdot \:49))/(2\cdot \:4) \\ By\text{ simplifying the numerator,} \\ √(14^2-4*4*49)=√(196-784)=√(-588)=14√(3)i \end{gathered}`

By substitution,

`\begin{gathered} x_(1,\:2)=(-14\pm \:14√(3)i)/(2\cdot \:4) \\ \mathrm{Separate\:the\:solutions} \\ x_1=(-14+14√(3)i)/(2\cdot \:4),\:x_2=(-14-14√(3)i)/(2\cdot \:4) \\ x_1=(-14+14√(3)i)/(2\cdot\:4)=-(7)/(4)+i(7√(3))/(4) \\ x_2=(-14-14√(3)i)/(2\cdot\:4)=-(7)/(4)-i(7√(3))/(4) \end{gathered}`

Hence, the solutions to the statement are:

`(7)/(2),-(7)/(4)+i(7√(3))/(4),-(7)/(4)-i(7√(3))/(4)`