Question

Solve the equation algebraically. Check the solutions graphically. $$ x^{2}-4=12 $$

Short answer

The solutions to the equation \(x^{2}-4=12\) are \(x = 4\) and \(x = -4\).

Step-by-step solution

Simplify the Equation

To begin solving this quadratic equation, it should first be rewritten in an easier to solve format. Subtract the constant 4 from 12 to get 8 and write it on the RHS: \[x^{2} - 4 - 12 = 0 \rightarrow x^{2} - 16 = 0 \]

Finding the Roots

Applying the square root property, which is whenever you have a variable squared equal to a number, your solutions are going to be plus or minus the square root of that number. Hence, the roots of the equation are \[x=± \sqrt{16}\] Thus, the solutions are \(x = 4\) and \(x = -4\).

Graphical Confirmation

To confirm graphically, plot the function \(f(x)= x^{2}-16\) against 'x' values. The graph is a parabola and it intersects the x-axis at points \(x = 4\) and \(x = -4\), verifying that these are indeed the solutions to the equation.