To graph the functions, make a table of values for each. Be aware that the radicand cannot be negative!
y=x+5 | y=x+3 | |||
---|---|---|---|---|
x | x+5 | y | x+3 | y |
-5 | -5+5 | 0 | -5+3 | -2 |
-4 | -4+5 | 1 | -4+3 | -1 |
-1 | -1+5 | 2 | -1+3 | 2 |
4 | 4+5 | 3 | 4+3 | 7 |
Now the points for each function obtained in the table will be plotted on the same coordinate plane. Then, the points of the linear function will be connected with a straight line, and the points of the radical function will be connected with a smooth curve.
The x-coordinate of the point of intersection of the functions gives the solution of the original equation.
The curve and the line intersect at (-1,2). This means that the solution to the equation is x=-1.
Next, we can identify the points that have the y-coordinate -1, then find the corresponding x-coordinates.
x=3
Multiply
Add and subtract terms
Calculate root
Radical equations can be solved algebraically using inverse operations. Specifically, to undo the radical, both sides of the equation can be raised to the same power as the index of the radical. For example,
(a−b)2=a2−2ab+b2
Calculate power and product
LHS−2x=RHS−2x
LHS+1=RHS+1
Rearrange equation
Use the Quadratic Formula: a=1,b=-6,c=5
-(-a)=a
Calculate power and product
Subtract term
Calculate root
State solutions
Add and subtract terms
Calculate quotient
x=5
Multiply
Subtract terms
Calculate root
Add terms
Since x=1 makes a true statement, it is a solution to the radical equation. With regard to discussing solutions, it can be said that 2=x+2x−1 has one solution and one extraneous solution.
γ=1.25
LHS⋅1−β2=RHS⋅1−β2
LHS/1.25=RHS/1.25
Use a calculator
LHS2=RHS2
(a)2=a
Calculate power
LHS−1=RHS−1
LHS/(-1)=RHS/(-1)
LHS=RHS
β>0
Calculate root
β=0.6
Calculate power
Subtract term
Calculate root
Use a calculator