In this explainer, we will learn how to write a quadratic equation given the roots of another quadratic equation.
Consider a quadratic equation in the form , where , , and are constants, and is not equal to 0. The quadratic formula states that the solutions to, or roots of, the equation are
Sometimes, we refer to these roots as and . Algebraically, we can show that , or the sum of these two roots, is equal to , and that , or the product of these two roots, is equal to , as follows:
This allows us to make a generalization about the relationship between a quadratic equation and its roots.
Property: Relationship between a Quadratic Equation and Its Roots
For any quadratic equation in the form , the sum of the roots and is equal to , and the product of the roots is equal to . That is,
Notice that if , or the leading coefficient in the equation , is equal to 1, then the sum of the roots becomes and the product of the roots becomes
We can, therefore, make another generalization about the relationship between a quadratic equation and its roots when the leading coefficient of the equation is 1.
Property: Relationship between a Quadratic Equation with a Leading Coefficient of 1 and Its Roots
For any quadratic equation in the form for which , the sum of the roots and is equal to , or the negative of the coefficient of , and the product of the roots is equal to , or the constant term. That is,
Using these relationships, we are able to form a quadratic equation using another quadratic equation and its roots. For example, if we are given that the roots of the equation are and , we know that the sum of these roots is equal to , or 7 and that the product of these roots is equal to 10. With this knowledge, we can form equations that we can use to help determine the quadratic equation whose roots are and . Letβs look at how we would solve problems such as this in the examples that follow.
Example 1: Forming a Quadratic Equation Using Another Quadratic Equation and Its Roots
Given that and are the roots of the equation , find, in its simplest form, the quadratic equation whose roots are and .
Answer
We know that for any quadratic equation in the form for which , the sum of the roots is equal to , or the negative of the coefficient of , and the product of the roots is equal to , or the constant term.
In the quadratic equation , we can see that the value of is 1, the value of is 8, and the value of is 12. This means that since the roots of this equation are and , we know that
The problem asks us to find another quadratic equation in its simplest form whose roots are and . Suppose this other quadratic equation is in the form . If we assume that , then we know that must be equal to and that must be equal to . First, letβs use the equation to help find a value for the expression :
Since the value of is , we know that this is the value of , and hence, the value of must be 14.
Next, letβs use the equation to help find a value for the expression :
Notice that in the second term on the left side of the equation, we have the expression being multiplied by 3. We already know that the value of the expression is , so we can now substitute for and then isolate :
Since is equal to 45, we know that this is the value of . Therefore, since , , and , we have shown that the quadratic equation has the roots and .
Now, letβs look at a similar example in which we must form a quadratic equation using another quadratic equation and its roots.
Example 2: Forming a Quadratic Equation Using Another Quadratic Equation and Its Roots
Given that and are the roots of the equation , find, in its simplest form, the quadratic equation whose roots are and .
Answer
Recall that for any quadratic equation in the form for which , the sum of the roots is equal to , or the negative of the coefficient of , and the product of the roots is equal to , or the constant term.
We can see that, in the quadratic equation , the value of is 1, the value of is , and the value of is 5. We were told that the roots of this equation are and , so we know that
The problem asks us to find another quadratic equation in its simplest form whose roots are and . Suppose this other quadratic equation is in the form . If we assume that , then we know that must be equal to and that must be equal to . In order to find a value for the expression , we must begin by squaring both sides of the equation and then expanding the left side:
Notice that in the second term on the left side of the equation, we have the expression being multiplied by 2. Since we already know that the value of the expression is 5, we can now substitute 5 for and then isolate :
Since the value of is , we know that this is the value of , and hence, the value of must be 6.
Now, letβs find a value for the expression . We can do so by squaring both sides of the equation and then distributing the exponent:
Since is equal to 25, we know that this is the value of . Therefore, since , , and , we have shown that the quadratic equation has the roots and .
Next, we will again form a quadratic equation using another quadratic equation and its roots. However, this time, the coefficients we arrive at will not be integers, so we will have to multiply both sides of the equation by a constant to eliminate the fractions.
Example 3: Forming a Quadratic Equation Using Another Quadratic Equation and Its Roots
Given that and are roots of the equation , find, in its simplest form, the quadratic equation whose roots are and .
Answer
Recall that for any quadratic equation of the form for which , the sum of the roots is equal to (the negative of the -coefficient), and the product of the roots is equal to (the constant term).
Now, the given equation has an -coefficient of , so rather than 1, but we can convert it into the desired form by multiplying the entire equation by . This gives us
From this equation, we have a value for of and for of 9, so since the roots are and , we have
Now the problem asks us to find a quadratic equation with roots and . Using the above result, if the equation we want to find is of the form (where ), then the sum of the two roots must equal , and the product must equal . To simplify the expression for the sum, we can multiply the numerator and denominator of each fraction by the denominator of the other and simplify to get
Notice that the numerator of the resulting expression is the sum of the roots of the original equation, and the denominator of the expression is the product of the roots. We have already determined that the sum of the roots is equal to 7, and the product is equal to 9, so this tells us , which means .
For the product, recall that we have . This simplifies by multiplying the denominators together to get
Since we know , this means . Now, taking the values we have calculated for and , we have the following equation:
As a final step, we can simplify this by multiplying the equation by 9 to make the coefficients into integers. This gives us
Having seen the approach we can take for a problem where the roots are fractions, let us once more apply this method to a similar but slightly more complex problem.
Example 4: Forming a Quadratic Equation Using Another Quadratic Equation and Its Roots
Given that and are the roots of the equation , find, in its simplest form, the quadratic equation whose roots are and .
Answer
Remember that for any quadratic equation in the form for which , the sum of the roots is equal to , or the negative of the coefficient of , and the product of the roots is equal to , or the constant term.
In the quadratic equation , we can see that the value of is 1, the value of is , and the value of is 12. Thus, since the roots of this equation are and , we know that
The problem asks us to find another quadratic equation in its simplest form whose roots are and . Suppose this other quadratic equation is in the form . If we assume that , then we know that must be equal to and that must be equal to .
In order to find a value for the expression , we must begin by finding a common denominator for the two fractions in the expression and then adding the fractions together to get , or .
Now, letβs square both sides of the equation and then expand the left side:
Notice that in the second term on the left side of the equation, we have the expression being multiplied by 2. Since we already know that the value of the expression is 12, we can substitute 12 for and then isolate :
Next, letβs find a value for the expression . We can do so by squaring both sides of the equation and then distributing the exponent:
Now, substituting and into the expression , we get for , or for , the coefficient of in the equation we are seeking.
Also, substituting into the expression , we get for , the constant term in the equation. This gives us an equation of .
Finally, to eliminate the fractions, we can multiply both sides by 144 to get , or , for our equation.
Now, letβs look at an example of how to form a quadratic equation using a nonmonic quadratic equation and its roots. A nonmonic quadratic equation has a leading coefficient that is not 1.
Example 5: Forming a Quadratic Equation Using a Nonmonic Quadratic Equation and Its Roots
Given that and are the roots of the equation , find, in its simplest form, the quadratic equation whose roots are and .
Answer
We know that for any quadratic equation in the form , the sum of the roots is equal to , and the product of the roots is equal to . In the equation , the value of is 3, the value of is 16, and the value of is . Thus, since the roots of this equation are and , we know that
The problem asks us to find another quadratic equation in its simplest form whose roots are and . Suppose this other quadratic equation is in the form . Remember that for any quadratic equation in the form for which , the sum of the roots is equal to , or the negative of the coefficient of , and the product of the roots is equal to , or the constant term.
Thus, if we assume that , then we know that must be equal to , or and that must be equal to , or .
Substituting into the expression , we get for , or for , the coefficient of in the equation we are seeking.
Also, substituting into the expression , we get for , the constant term in the equation. This gives us an equation of .
Finally, to eliminate the fractions, we can multiply both sides by 12 to get , or , for our equation.
Finally, we will work on another problem in which we must form a quadratic equation using a nonmonic quadratic equation and its roots.
Example 6: Forming Quadratic Equations in the Simplest Form Using the Relation between a Quadratic Equation and Its Roots
If and are the roots of the equation , find, in its simplest form, the quadratic equation whose roots are and .
Answer
We know that for any quadratic equation in the form , the sum of the roots is equal to , and the product of the roots is equal to . In the equation , the value of is 2, the value of is , and the value of is 1. Thus, since the roots of this equation are and , we know that
The problem asks us to find another quadratic equation in its simplest form whose roots are and . Suppose this other quadratic equation is in the form . Recall that, for any quadratic equation in the form for which , the sum of the roots is equal to , or the negative of the coefficient of , and the product of the roots is equal to , or the constant term.
Therefore, if we assume that , then we know that must be equal to and that must be equal to , or .
In order to determine a value for the expression , letβs start by squaring both sides of the equation and then expanding the left side:
Notice that in the second term on the left side of the equation, we have the expression being multiplied by 2. Since we already know that the value of the expression is , we can now substitute for and then isolate :
Next, multiplying both sides of the equation by 2, we get
Since the value of is , we know that this is the value of , and hence, the value of must be .
Now, letβs determine a value for the expression . We can begin by squaring both sides of the equation and then distributing the exponent:
Next, multiplying both sides of the equation by 4, we get
Since the value of is 1, we know that this is the value of .
Thus, we can now substitute , , and into to get the equation . Finally, to eliminate the fraction, we can multiply both sides by 2 to get , or , for our equation.
Now letβs finish by recapping some key points.
Key Points
- For any quadratic equation in the form , the sum of the roots is equal to , and the product of the roots is equal to .
- For any quadratic equation in the form for which , the sum of the roots is equal to , or the negative of the coefficient of , and the product of the roots is equal to , or the constant term.
- It is possible to form a quadratic equation using another quadratic equation and its roots.
- A nonmonic quadratic equation is a quadratic equation with a leading coefficient that is not 1.
