Question

Solve the equation -x² + 1 = x − 1 graphically. Your answer must include:

• A hand-drawn or calculator-created image of the graph with the equations f(x) and g(x) identified, along with labeled axes and any point(s) of intersection
• A statement listing the solution(s) to the equation
• Your work verifying that the solution(s) create a true statement in the original equation

Answer 1

• The graph of the two functions is shown below:

• From the graph, the solution of the graph is the value of $x$ where the two $f(x) \text{ and } g(x)$ intersect. The value of $x$ at these points are: $1 \text{ and } -2$

• To verify the solution, we solve the quadratic equation by factorisation:

$\text{Solving the equation step-by-step. }\\ -x^{2}+1=x-1\\[2mm] \text{ Step 1: Subtract }x-1 \text{ from both sides.}\\[1mm] \begin{aligned} &-x^{2}+1-(x-1)=x-1-(x-1) \\ &-x^{2}-x+2=0 \end{aligned}\\[3mm] \text{Step 2: Factor left side of equation.}\\ (-x+1)(x+2)=0\\[3mm] \text{Step 3: Set factors equal to } 0 .\\ -x+1=0 \text { or } x+2=0\\[2mm] x=1 \text { or } x=-2$