Solving Quadratic Equations with Square Roots

Let's review how can we find the solutions to an equation of the form $x^2=d$ by using square roots. For this, we can take the square root of each side of the equation to isolate $x.$ Notice that there are three possible cases according to the value of $d.$

• Case $\mathbf{d}>0.$ In this case, since $\sqrt{d}\ne \text{-}\sqrt{d},$ the equation $x^2=d$ has two real solutions, $x=\pm \sqrt{d}.$
• Case $\mathbf{d}=0.$ In this case, since $\sqrt{0}=\text{-}\sqrt{0},$ the equation $x^2=d$ has one real solution, $x=0.$
• Case $\mathbf{d}<0.$ In this case, since $\sqrt{d}$ is not a real number, the equation $x^2=d$ has no real solutions.

Thus, by studying the value of $d$ the number of solutions can be found.