In this explainer, we will learn how to recognize the relationship between the coefficients of a quadratic equation and its roots.
We begin with a brief review of how to factor a quadratic equation to find its roots. Consider the equation
The quadratic expression on the left-hand side of this equation can be factored as which leads to the roots and . When we look more closely at this factorization process, we first looked for a pair of numbers that had a product of the constant term 10 and sum to the coefficient . After factoring, this pair of numbers turned out to be the negative of the roots of the quadratic equation. Thinking about this process in reverse, the roots of this equation, and (sometimes referred to as and ), should satisfy
In other words, we are starting with the presumption that the coefficients of the quadratic equation contain information about the roots. We can verify that this is true in this example, since we know that the roots are and , and
But is this true for any quadratic equation? What is the precise relationship between the coefficients of any quadratic equation and its roots?
Letβs start with a simple quadratic equation, , whose leading coefficient (i.e., the coefficient of ) is 1. Say that this equation factors into for some and . Then, the quadratic equation has roots and . We distribute this equation across the parentheses:
The middle two terms have in common, so we can take out this common factor to write
After the expansion is complete, we can compare the coefficients of this equation with those of the original equation, . Comparing this equation with the original quadratic equation, , we note that
In other words, quadratic equations of the form with roots and must satisfy
So, what can we say about the coefficients in a more general quadratic equation, where the leading coefficient is not equal to 1? Consider a quadratic equation with roots and . Since , We can divide both sides by to write
Now, we have a quadratic equation with a leading coefficient of 1. Then, using our previous results, the coefficients and the roots of the quadratic equation must satisfy the following equations.
Theorem: Coefficients and Roots of a Quadratic Equation
Let be a quadratic equation with roots and . Then, the roots must satisfy
For simpler quadratic equations in the form with roots and , we have the abbreviated formulae
These formulae stand true for all quadratic equations, even when the roots are complex valued or are repeated. The same formulae can be recovered using the quadratic formula.
For example, consider the quadratic equation . Without solving the equation, we can find the sum and product of its roots. Since we have where and are the roots of this quadratic equation.
We can verify this by computing the roots using the quadratic roots formula. We have
So, we have
Then, which agrees with what we obtained above using the theorem.
Also, using the difference of squares formula, , we can compute which also agrees with our result using the theorem.
In our first example, we will demonstrate how our theorem can help us find the sum of the roots without solving the equation.
Example 1: Relation between the Coefficients of a Quadratic Equation and Its Roots
Without solving the equation , find the sum of its roots.
Answer
We recall that, for quadratic equations with roots and ,
Since the given quadratic equation is , we have
Using the formula above, the sum of its roots is equal to
So, the sum of its roots is equal to .
Using the relationship between the coefficients and the roots of quadratic equations, we can find quadratic equations given their roots. This is the reverse process to problems where we find the roots of a quadratic equation. For these types of problems, we can also verify our answer by finding the roots of the quadratic equation and comparing them with the given roots.
When deriving a quadratic equation from the roots, it is easier to start with the simpler form where the leading coefficient is equal to 1. If either of the coefficients and is a fraction, then we can multiply both sides of the equation by the common denominator to further simplify the quadratic equation.
For example, let us derive a quadratic equation with roots and . Starting with the quadratic equation , we can use our theorem to write and
So, we get and , leading to the quadratic equation
Since this equation contains a noninteger coefficient, , we can further simplify this equation by multiplying both sides by 2. Then, we get
So, the simplest quadratic equation with roots and is .
We can verify our answer by solving the quadratic equation by factoring. We can factor the quadratic expression as , so
This leads to the roots and , which agrees with the given roots. This verifies our answer.
We will now consider a few examples where we derive the quadratic equation from the roots.
Example 2: The Relation between the Coefficient of a Quadratic Equation and Its Roots
Given that and are the solutions of the equation , find the values of and .
Answer
We recall that, for quadratic equations of the form with roots and ,
We are given that and are roots of the equation, so , which leads to . Also, we have , leading to .
We verify our answer by working backward to solve the quadratic equation by factoring. Our answer leads to the quadratic equation
By factoring we can obtain the roots and . These roots agree with the ones given by the problem.
So, and .
Example 3: Forming Quadratic Equations in the Simplest Form given Their Roots
Find, in its simplest form, the quadratic equation whose roots are and .
Answer
Since we are looking for the simplest form of a quadratic equation, we should look for a quadratic equation with integer coefficients. Let us start with the form . If any of the coefficients are fractions, then we can multiply both sides of the equation by the common denominator to simplify the equation.
We remember that if and are the roots of this equation, then
We are given that and are its roots, so , leading to . Also, we have , leading to . Since the coefficients and are integers, we do not need to alter this equation further. So we obtain the quadratic equation .
We can verify our answer by solving the quadratic equation by factoring: so we obtain the roots and . These roots agree with the ones given in the problem.
So the quadratic equation with roots and , in its simplest form, is .
Example 4: Forming a Quadratic Equation in the Simplest Form given Its Roots
Find, in its simplest form, the quadratic equation whose roots are and .
Answer
Let us start with the form . If any of the coefficients are fractions, then we can multiply both sides of the equation by the common denominator to simplify the equation. We remember that if and are the roots of this equation, then
We are given that and are its roots, so which leads to . Also, recalling the difference of squares formula, , we get
So we obtain the quadratic equation .
We can verify our answer by solving the quadratic equation using the quadratic formula:
These roots agree with the ones given in the problem.
So the quadratic equation with roots and , in its simplest form, is .
In our next example, we will demonstrate how the procedure holds for complex roots.
Example 5: Identifying Quadratic Equations with a Given Pair of Complex Roots
Which quadratic equation has roots ?
Answer
Since all the given choices carry 1 as the leading coefficient, we start with the quadratic equation . We remember that if and are the roots of this equation, then
We are given that and are its roots, so which leads to . Also, recalling the difference of squares formula, , we get
We obtain the quadratic equation .
We can verify our answer by solving the quadratic equation using the quadratic formula:
These roots agree with the ones given in the problem.
So the quadratic equation with roots and , in its simplest form, is .
In the last two examples, we will look at quadratic equations containing an unknown. We will use the relationship between the coefficients and the roots of quadratic equations to solve these problems.
Example 6: Finding the Value of an Unknown in a Quadratic Equation given One of Its Roots
Given that is a root of the equation , determine the value of .
Answer
We begin by putting the given equation in the standard form,
We recall that, given a quadratic equation , is equal to the product of its roots. In our example, . We are given one of its roots, , but we do not know the other root, which we will denote .
Then,
So the second root of this equation is .
We also recall that, given a quadratic equation , is equal to the sum of its roots. In our example, . Since we know that the roots are and , then
So, .
Example 7: Finding the Value of an Algebraic Expression Using the Relation between the Coefficients of a Quadratic Equation and Its Roots
If and are the roots of the equation , what is the value of ?
Answer
There are two ways to approach this problem. The first is to use the relation between the coefficients and the roots of a quadratic equation to reproduce the required expression. The second is to find the roots of the quadratic equation and then to compute the required expression. We begin with the first approach, and we will follow with the second one.
We recall that, given a quadratic equation with roots and ,
In our example, we note that and , so
Now, we observe that the required expression can be written as
Since we know and ,
So, , following the first method.
Now, let us examine the second method. Given the equation , we can factor the left-hand side, which gives us the roots and . We can assign and (the result would be the same if we assigned and , because the required expression is and addition is commutative).
We compute the required expression:
So is equal to 82.
Key Points
- Coefficients of a quadratic equation contain information about the sum and the product of its roots.
- For quadratic equations , if and are the roots, then
- For general quadratic equations , if and are the roots, then
- When deriving a quadratic equation from given roots and , we can use the abbreviated formulae