Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use other methods for solving a quadratic equation. One method of solving quadratic equation is known as completing the square.
Using this process, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, [latex]a[/latex], must equal [latex]1[/latex]. If it does not, then divide the entire equation by [latex]a[/latex]. Then, we can use the following procedures to solve a quadratic equation by completing the square.
completing the square
The goal of completing the square is to transform a quadratic equation of the form [latex]ax^2+bx+c = 0[/latex] into a perfect square trinomial:
[latex](x-h)^2 = k[/latex],
which can easily be solved by taking square roots.
How to: Solve a quadratic equation by completing the square
Rearrange your equation so that it is in standard form: [latex]ax^2 + bx + c = 0[/latex]. Divide by [latex]a[/latex] if [latex]a \neq 1[/latex].
Isolate the constant term by moving it to the right side of the equation.
Add [latex](\frac{b}{2})^2[/latex] on both sides of the equation.
Form the perfect square trinomial.
Solve for [latex]x[/latex] using the square root property.
Perfect Square Trinomials:
[latex]a^2 + 2ab + b^2 = (a + b)^2[/latex]
[latex]a^2 - 2ab + b^2 = (a - b)^2[/latex]
We will use the example [latex]{x}^{2}+4x+1=0[/latex] to illustrate each step.
Given a quadratic equation that cannot be factored and with [latex]a=1[/latex], first add or subtract the constant term to the right sign of the equal sign.
[latex]{x}^{2}+4x=-1[/latex]
Multiply the b term by [latex]\frac{1}{2}[/latex] and square it.
The solutions are [latex]x=-2+\sqrt{3}[/latex], [latex]x=-2-\sqrt{3}[/latex].
Remember that we are permitted, by the properties of equality, to add, subtract, multiply, or divide the same amount to both sides of an equation. Doing so won’t change the value of the equation but it will enable us to isolate the variable on one side (that is, to solve the equation for the variable).The square root property gives us another operation we can do to both sides of an equation, taking the square root. We just have to remember when taking the square root (or any even root, as we’ll see later), to consider both the positive and negative possibilities of the constant.Solve the quadratic equation by completing the square:
[latex]{x}^{2}-3x - 5=0[/latex]
First, move the constant term to the right side of the equal sign by adding [latex]5[/latex] to both sides of the equation.
[latex]{x}^{2}-3x=5[/latex]
Then, take [latex]\frac{1}{2}[/latex] of the [latex]b[/latex] term and square it.
The solutions are [latex]x=\frac{3}{2}+\frac{\sqrt{29}}{2}[/latex], [latex]x=\frac{3}{2}-\frac{\sqrt{29}}{2}[/latex].
Solve the quadratic equation using completing the square:
[latex]3x^2-6x-9=0[/latex]
[latex]\begin{align*} \text{Divide by the coefficient of } x^2: & \quad \frac{3x^2 - 6x - 9}{3} = 0 \\ & \quad x^2 - 2x - 3 = 0 \\ \text{Isolate the constant term:} & \quad x^2 - 2x = 3 \\ \text{Add } \left(\frac{-2}{2}\right)^2 = (-1)^2 = 1 \text{ on both sides}: & \quad x^2 - 2x + 1 = 3 +1 \\ \text{Form the perfect square trinomial:} & \quad (x - 1)^2 = 3 +1 \\ \text{Simplify and solve:} & \quad (x - 1)^2 = 4 \\ & \quad x - 1 = \pm 2 \\ & \quad x = 1 \pm 2 \\ & \quad x = 1 + 2 = 3 \text{ and } x = 1 - 2 = -1 \end{align*}[/latex]
Note that when solving a quadratic by completing the square, a negative value will sometimes arise under the square root symbol. Later, we’ll see that this value can be represented by a complex number (as shown in the video help for the problem below). We may also treat this type of solution as unreal, stating that no real solutions exist for this equation, by writing DNE.