Question

Topic: Nonlinear Systems of EquationsCreate a system of equations that includes one linear equation and one quadratic equation.Show all work in solving your system of equations algebraically.

Answer 1

• Equations

`y=x+2`
`y=x^2`

• Solutions (2, 4) and (–1, 1)

Step-by-step explanation

System of equations

• Linear equation

`y=x+2`

• Quadratic equation

`y=x^2`

We have to set both equations to 0 and equalize them:

• 1. Setting them to 0:

`0=x+2`
`0=x^2`

• 2. Equalizing them:

`x+2=x^2`

To solve the system using the General Quadratic Formula, we have to set the equation in the form ax² + bx+ c = 0:

`x^2-x-2=0`

Thus, in this case a = 1, b = -1 and c = -2. Using the formula:

`x_(1,2)=(-(-1)\pm√((-1)^2-4(1)(-2)))/(2(1))`
`x_(1,2)=(1\pm√(1+8))/(2)`
`x_(1,2)=(1\pm√(9))/(2)`

Finding both solutions:

`x_1=(1+3)/(2)=(4)/(2)=2`
`x_2=(1-3)/(2)=(-2)/(2)=-1`

Finally, replacing these values in the linear equation to find y:

`y_1=2+2=4`
`y_2=-1+2=1`

Therefore, the solutions are (2, 4) and (–1, 1).