In this explainer, we will learn how to solve quadratic equations using the quadratic formula.
Recall that a quadratic equation is an equation with one variable, where the highest order of any term is 2. This is made more explicit in the definition below.
Definition: Quadratic Equation
The equation , , with variable and constants , , and , is called a quadratic equation (or second-degree equation).
To solve quadratic equations, we can use the following methods:
- Factoring
- Completing the square
- Using the quadratic formula
- Solving graphically
So far, we have met factoring and completing the square. We will use completing the square to derive the quadratic formula, which is the method we will be focusing on in this explainer.
If we have some quadratic equation of the form , , with variable and constants , , and , then using completing the square, we can rearrange and solve for as follows.
First, we will divide through by , the coefficient of :
Next, we will subtract from both sides:
Then, to make a perfect square, we will add to both sides:
Since the left-hand side of the equation is in the form , a perfect square, then we can write this as :
Next, we will rearrange to get as the subject. First, we square root both sides (taking both the positive and negative square roots):
Next, we isolate :
We then simplify further by expanding the brackets in the radical and rearranging slightly:
We can now write the whole expression in the radical over a common denominator:
Since the denominator in the radical is a perfect square, then we can square root this and write it outside of the radical:
Combining the two terms, since they have the same denominator, we then get
So in its final form, is the quadratic formula, which is used for finding solutions of quadratic equations of the form , . This is stated in the definition below.
Definition: The Quadratic Formula
To solve a quadratic equation in the form , , with variable and constants , , and we can use the quadratic formula to solve for :
Whenever we want to use the quadratic formula, we need to ensure that our quadratic equation is equal to zero, with it in expanded form and simplified as much as possible, so that it is in the form , . Then, we need to identify what , , and each are. Following that, we can substitute into the quadratic formula and solve for . There are usually two values of , corresponding to the positive and negative square roots of , but sometimes there is only one, or even no real solutions, depending on the values of , , and .
We will discuss how to solve a quadratic equation which is already in the form , in our first example.
Example 1: Solving Quadratic Equations Using the Quadratic Formula
Find the solution set of the equation , giving values to two decimal places.
Answer
As the equation is a quadratic equation, then we can use one of the methods for solving quadratics. In this case, we are going to use the quadratic formula in order to solve. Recall, that for a quadratic equation in the form , , then
Since is already in the same form as since it is equal to zero, fully simplified, and written in descending powers of , then we can identify the values of , , and :
Substituting these values into the quadratic formula and solving for , we get
As we are asked to give the solution set to 2 decimal places, then evaluating , we get
So, the solution set of the equation is correct to 2 decimal places.
In the next example, we will consider how to solve a quadratic equation that is not in the form initially but, by rearranging, can be put in this form and then solved using the quadratic formula.
Example 2: Solving Quadratic Equations Using the Quadratic Formula
Find the solution set of in , giving values to two decimal places.
Answer
As the equation contains a term with , then this is likely to be a quadratic equation. To check, we can simplify by expanding brackets and making the equation equal to zero, as follows:
As the highest power in the equation is 2, then we can see this is a quadratic equation. Since it is now written in the form , then we can apply the quadratic formula, which states
We can see that for , , , and . Substituting this into the quadratic formula, we get
Simplifying, we get
As we are asked to give the solution set to 2 decimal places, then evaluating , we get
So, the solution set of correct to 2 decimal places is .
We can use the solutions of quadratic equations to find unknown parts of the equation, such as coefficients or constants. We can do this by substituting known and unknown parts into the quadratic formula and solving for the unknown part. We will explore how to do this in the next example.
Example 3: Finding Unknowns in Quadratic Equations Using the Quadratic Formula
Given that is a root of the equation , find the set of possible values of .
Answer
Since is of the form of a quadratic equation, then we can use the quadratic formula to find the values of .
The quadratic formula states that for an equation in the form , then
We can see by comparing with that , , and . Since we also know that one of the roots of the equation is then we can substitute , , and into the quadratic formula:
Simplifying, we get
We can simplify further by factoring out 4 and putting it outside the radical:
Next, we need to solve for . Since part of the equation contains , then once rearranged, it is likely to be the form of a quadratic equation. As such, we want to rearrange so itโs in the form , , so that we can apply the quadratic formula again (but this time with different values for , , and ).
Rearranging, we get
Now that the equation is in the form , , we can find , , and . By comparing, we can see that , , and . Substituting into the quadratic formula, we get
Simplifying, we get
Therefore, the possible values of are .
In addition to quadratic equations that contain quadratic terms, we can have equations that may not appear to be quadratic on first inspection but with some rearranging become quadratics. For example, can be rearranged to give , which is a quadratic equation. Therefore, we can solve equations, that once rearranged, become quadratics, and can do so using the quadratic formula. In our next example we will explore how to do this.
Example 4: Rearranging Equation to Solve Using the Quadratic Formula
Find the solution set of the equation in , giving values to one decimal place.
Answer
In order to solve , it is helpful to remove any variables in the denominators first. To do this we need to multiply through by the highest power of in the denominator, which is . Doing so gives us
As the highest power of the equation is 2, then we can see this is a quadratic equation. To solve using the quadratic formula, we need to rearrange to make this in the form , . In this case, it is helpful to move the terms on the left-hand side to the right-hand side so that the coefficients are positive (but it does not necessarily matter) so that it is easier to do calculations later:
Now that this is in the form , we can apply the quadratic formula, which states
Comparing with , we can see that , , and . Substituting into the quadratic formula, we get
Simplifying, we get
As the question requires us to find the solution set accurate to one decimal place, then we will evaluate this, giving us
Therefore, the solution set for the equation correct to one decimal place is .
In addition to equations that can be rearranged to give a quadratic equation, we can have equations that are not quadratics themselves but can be solved using the quadratic formula as they are in a quadratic form. For example, is a quartic equation, as its highest power is 4, but as it takes the form of , , where the variable represents or , then it can be solved using the quadratic formula since it has a quadratic form.
In the next example, we will consider how to solve a quartic equation by writing it in the form of a quadratic equation and using the quadratic formula.
Example 5: Using the Quadratic Formula to Solve a Quartic Equation
Using the quadratic formula, find all the solutions to .
Answer
To find the solutions of the equation using the quadratic formula, we need to write the equation in the form of a quadratic equation. We can see that we have even powers of , meaning we can replace with another variable, say , giving us . Doing so gives us
We can now see that takes the form of a quadratic equation , . We can then apply the quadratic formula to solve for , which states
Since , , and , then substituting gives us
Simplifying, we then get
Remember that we let , so
Square rooting both sides and solving for gives us
Therefore, all solutions to the equation are
In this explainer, we have discussed how to solve quadratic equations using the quadratic formula. In order to get equations into the form required, we have either rearranged the equation, or substituted to make it into a quadratic form. Letโs recap the key points.
Key Points
- We can solve quadratic equations of the form , using the quadratic formula
- By rearranging, some equations can be written as a quadratic equations of the form , , and then solved using the quadratic formula.
- Some equations that are not quadratic equations themselves, can be solved using the quadratic formula if they take the form , , where can represent another function.
