Quadratic equations can be solved using many methods. You should already be familiar with factoring to solve some quadratic equations. However, not all quadratic equations can be factored. In this section, you will learn two other ways to solve quadratic equations.

Solve a Quadratic Equation by the Square Root Property

One way to solve the quadratic equation [latex]x^{2}=9[/latex] is to subtract [latex]9[/latex] from both sides to get right side of the equation equal to 0.

[latex]x^{2}-9=0[/latex]

The expression on the left can be factored; it is a difference of squares.

[latex]\left(x+3\right)\left(x–3\right)=0[/latex].

Using the zero-product property, [latex]x+3=0[/latex] or [latex]x –3=0[/latex], so [latex]x=−3[/latex] or [latex]3[/latex].

Another property that would let you solve this equation more easily is called the square root property.

When using the Square Root Property to solve an equation such as the one above ($$x^2=p$$), the solutions can be abbreviated as $$x=\pm \sqrt{p}$$. This is read as “x equals plus or minus the square root of p.” These still represent the same solutions of [latex]x=\sqrt{p}[/latex] or [latex]x=-\sqrt{p}[/latex].

Notice that there is a difference here in solving [latex]x^{2}=9[/latex] and simplifying [latex]\sqrt{9}[/latex]. For [latex]x^{2}=9[/latex], you are looking for all numbers whose square is [latex]9[/latex]. For [latex]\sqrt{9}[/latex], you only want the principal (nonnegative) square root.

In the example above, you can take the square root of both sides easily because there is only one term on each side. In some equations, you may need to do some work to get the equation in this form. You will find that this involves isolating [latex]x^{2}[/latex].

In our first video, we will show more examples of using the square root property to solve a quadratic equation.

In the next example, we have a quantity squared and that is what we need to isolate to solve this using the Square Root Property.

In the next video, you will see more examples of using square roots to solve quadratic equations.

Solve a Quadratic Equation by Completing the Square

When you square a number, the product is called a perfect square. For example, $$5^2=5\cdot 5=25$$ so 25 is a perfect square. Here is a list of some integers that are also perfect squares: $$1,4,9,16,25,36,49,64,81,100…$$. We can also visualize integers that are perfect squares by arranging equal rows and columns to form a square, as shown below.

Take a closer look at the perfect square trinomials below. What do you notice about the constant terms? How are those constant terms related to the factored form and the binomial squared?

Did you notice in the first example that the constant term, $$25$$, of the trinomial is a perfect square? We know that $$25=5\cdot 5=5^2$$ and this helps us to confirm why there are $$5$$’s in the factored form and the $$5$$ in the binomial squared form. Did you notice that the constant term of the trinomial is always positive, even when the $$x$$ term is negative? Notice that this same idea applies to the other two examples above.

Let’s take a look at the $$x$$ term (middle term) of the trinomial. How is it related to the binomial that is being squared? Looking at the first example above, did you notice that if you find half of the coefficient of the middle term of the trinomial, $$10$$, that it is equal to $$5$$? Which is also the second term of the binomial squared. Does this same pattern hold true for the other above examples? Yes it does. In the second example, if we take half of $$-6$$, we get $$-3$$ and in the third example if we take half of $$-8$$ we get $$-4$$.

Not all trinomials can be factored and not all trinomials in the form of $$ax^2+bx+c$$ are perfect square trinomials. When they are not, we will need to change these trinomials into perfect square trinomials so that we can use this method to solve quadratic equations. Using this method, 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. In this course, we will only be completing the square on trinomials that have a leading coefficient of $$1$$. In later courses you will learn how to complete the square when the leading coefficient does not equal 1.

First, let’s make sure we remember how to factor a perfect square trinomial.

If this were an equation, we could solve using either the square root property or the zero product property. If you do not start with a perfect square trinomial, you can complete the square to make what you have into one.

In this course when we complete the square, the leading coefficient, a, will equal 1. Let’s look at an equation that does not have a perfect square trinomial.

$$x^2+8x-84=0$$

To change the left side of the equation into a perfect square trinomial, let’s start by adding 84 to both sides of the equation which is:

$$x^2+8x=84$$ and rewrite the equation to look like $$x^2+8x+\color{BurntOrange}{\_\_\_}\color{black}{= 84+}\color{BurntOrange}{\_\_\_}$$.

Now we need to figure out what to put in the blanks to make the left side of the equation a perfect square trinomial. To do this, we will find half of the coefficient of the middle term which is $$\dfrac{8}{2}=4$$. Then take that answer and square it $$(4)^2=16$$. So, $$16$$ is the number that we need to fill in the blanks with to make the left side into a perfect square trinomial.

$$\begin{align}x^2+8x+\color{BurntOrange}{16} &= 84+\color{BurntOrange}{16}\\ x^2+8x+16 &= 100\end{align}$$

Now, rewrite the left side of the equation as a binomial squared. Then square root both sides of the equation (using the Square Root Property) to solve for $$x$$.

$$\begin{align}(x+4)^2 &= 100\\ \sqrt{(x+4)^2} &= \sqrt{100}\\ x+4 &= \pm 10\end{align}$$

Now write as two equations and solve each equation by subtracting $$4$$ from both sides of each of the equations.

$$x+4=10\hspace{5mm}$$ OR $$\hspace{1cm}x+4=-10$$

$$x=6\hspace{1.5cm}$$ OR $$\hspace{1cm}x=-14$$

You might be wondering why this method is called Completing the Square. The following video will help you visualize geometrically what is meant by Completing the Square.

We can use the following steps to solve a quadratic equation by completing the square.

In the next video, you will see more examples of how to use completing the square to solve a quadratic equation.

You may have noticed that because you have to use both square roots, all the examples have two solutions. Here is another example that is slightly different.

Take a closer look at this problem and you may see something familiar. Instead of completing the square, try adding [latex]47[/latex] to both sides in the equation. The equation [latex]x^{2}+16x+17=−47[/latex] becomes [latex]x^{2}+16x+64=0[/latex]. Can you factor this equation using grouping? (Think of two numbers whose product is 64 and whose sum is [latex]16[/latex]).

It can be factored as [latex](x+8)(x+8)=0[/latex], of course! Knowing how to complete the square is very helpful, but it is not always the only way to solve an equation.

In our last video, we show an example of how to use completing the square to solve a quadratic equation whose solutions are irrational.

In the next example, take notice of the type and number of solutions.

Summary

Completing the square is used to change a binomial of the form [latex]x^{2}+bx[/latex] into a perfect square trinomial [latex]{{x}^{2}}+bx+{{\left( \frac{b}{2} \right)}^{2}}[/latex] which can be factored to [latex]{{\left( x+\frac{b}{2} \right)}^{2}}[/latex]. When we have a quadratic equation that cannot be factored, we can use completing the square to solve it. When solving quadratic equations by completing the square, be careful to add [latex]{{\left( \frac{b}{2} \right)}^{2}}[/latex] to both sides of the equation to maintain equality. The Square Root Property can then be used to solve for [latex]x[/latex]. With the Square Root Property, be careful to include both the principal square root and its opposite. Be sure to simplify as needed.