Question

Find and prove algebraically the solutions (coordinate points) to the system of equations?f(x)= 2x^2 + 4x + 1 and g(x) = 14x - 7(This is from the unit Linear and Quadratic equations. It's the only part of my homework I don't understand.)

Answer 1

Given

`\begin{gathered} f(x)=2x^2+4x+1_{} \\ g(x)=14x-7 \end{gathered}`

Let's equate f(x) and g(x)

`\begin{gathered} 2x^2+4x+1=14x-7 \\ \end{gathered}`

Add similar element

`\begin{gathered} 2x^2+4x+1=14x-7 \\ 2x^2+4x-14x+1+7=0 \\ 2x^2-10x+8=0 \end{gathered}`

We can solve by factorisation

`\begin{gathered} 2x^2-8x-2x+8=0 \\ (2x^2-8x)-(2x-8)=0 \\ 2x(x-4)-2(x-4)=0 \\ x-4=0 \\ 2x-2=0 \\ \end{gathered}`
`\begin{gathered} 2x-2=0 \\ \text{divide both sides by 2} \\ (2x)/(2)=(2)/(2) \\ x=1 \\ \\ \text{and } \\ x-4=0 \\ x=4 \end{gathered}`

Graphically

The final answer

`\begin{gathered} x=1 \\ x=4 \end{gathered}`