Question

Which of the following coordinates represents the X-coordinate of one of the points of intersection for the equations below?

Answer 1

Graphically, the point of intersection between these two lines is the point where they are exactly equal

From this fact, we can calculate the value of the coordinates that define it, formally, if we consider two straight lines expressed as follows

`x^2-4x+3=-0.5x+5`

Now we solve this first, we multiply by 10 both sides

`10x^2-40x+30=-5x+50`

Second, We subtract -50 from both sides

`\begin{gathered} 10x^2-40x+30-50=-5x+50-50 \\ 10x^2-40x-20=-5x \end{gathered}`

Third, we add +5x to both sides

`\begin{gathered} 10x^2-40x-20+5x=-5x+5x \\ 10x^2-35x-20=0 \end{gathered}`

Fourth, we solve for the quadratic formula, let's remember the quadratic formula

`\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

In this case, a=10, b=-35, and c=-20, we solve both roots

`\begin{gathered} \frac{-(-35)\pm\sqrt[]{-35^2-4\cdot10\cdot(-20)}}{2\cdot10} \\ \frac{35\pm\sqrt[]{1225+800}}{20} \\ \frac{35\pm\sqrt[]{2025}}{20} \\ (35\pm45)/(20) \end{gathered}`

For the negative root

`\begin{gathered} (35-45)/(20) \\ -(10)/(20) \\ -(1)/(2)=-0.5 \end{gathered}`

For the positive root

`\begin{gathered} (35+45)/(20) \\ (80)/(20) \\ (8)/(2)=4 \end{gathered}`

That is, the x coordinates where the two lines intersect are x=-0.5 and x=4

In conclusion, the answer in this case, is 4 since among the options it is the only one that coincides with a coordinate where both lines intersect