In this explainer, we will learn how to solve quadratic equations with no linear term using the square root property.
Definition: Quadratic Equation
A quadratic equation is an equation of the form where , , and are constants and .
When we solve a quadratic equation, we find the values of for which . A quadratic equation will have up to two different solutions for . In this explainer, we will look at how to solve a simple quadratic which has no linear term, that is, a quadratic equation in the form .
Let us recall the definition of real numbers and rational and irrational numbers.
Definition: Real Numbers
A real number is a positive or negative number, including those which have decimal places. A real number can be rational or irrational. We can write real numbers as belonging to the set .
Definition: Rational and Irrational Numbers
A rational number is a number which can be written in the form , where and are integers and . We can write rational numbers as belonging to the set .
An irrational number is one which cannot be written in the form , where and are integers.
To begin, let us consider the equation
If we wished to solve this for x, we would take the square root of both sides of the equation. We could solve as , so . However, since would also give an answer of 25, we can indicate the positive and negative roots here as
So,
It is important to remember that the use of indicates two solutions, so represents both and .
Let us now look at some examples of solving quadratic equations by taking square roots.
Example 1: Solving a Quadratic Equation by Taking Square Roots
Solve .
Answer
Starting from we take square roots of both sides of the equation. Recall that when taking square roots we need to consider both the positive and the negative roots. Therefore,
Subtracting 6 from both sides gives us
Therefore,
Hence, our answer is or .
Example 2: Solving a Quadratic Equation by Taking Square Roots
Find the solution set of the equation .
Answer
To begin solving this, we first multiply both sides of the equation by , which gives
Next, we multiply both sides by 9, giving us
Taking the square root of both sides, considering both the positive and the negative roots, we have
Therefore, and our answer is the set .
Example 3: Solving a Quadratic Equation by Taking Square Roots
Find the solution set of the equation in .
Answer
We perform the same operations to both sides of the equation to solve for . Taking the square root of both sides, considering both the positive and the negative roots, gives us
We can now add 5 to both sides to isolate , giving us
Therefore, the solution set is .
We will now look at an example where we first need to rearrange the equation to collect terms together, before solving using square roots.
Example 4: Solving a Quadratic Equation by Taking Square Roots
Determine the solution set of the equation .
Answer
To begin solving this equation, we collect all the like terms. Subtracting from both sides and then subtracting 15 from both sides give us
We can now divide both sides by , which gives us Taking the square root of both sides, considering both the positive and the negative roots, gives us Therefore, the solution set is .
Let us now look at an example where we can use the square root method to attempt to solve a quadratic but which gives no solutions for .
Example 5: Solving a Quadratic Equation by Taking Square Roots
Determine the solution set of the equation , given that .
Answer
We can solve this by performing the same operations to both sides of the equation to solve for .
Subtracting 9 from both sides of the equation and then dividing by 44 give us
We can now take the square root of both sides, considering both the positive and the negative roots, which gives us
However, neither nor gives us a real solution, since each takes the square root of a negative value. Therefore, the solution set of the equation is the null set: .
In this example, we correctly followed the square root method, but there are no real solutions for . If we consider the graph of the function , having no solutions for simply means that the graph would not pass through the -axis.
Key Points
- We can use the square root method to solve a quadratic equation when the quadratic has no terms in .
- We can use the square root method by following the steps below.
- Collect the -terms together, and collect the constant terms on the other side of the equation.
- Take the square root of both sides of the equation.
- It is important to remember that when taking a square rootwe must consider both the positive and the negative values of the roots. Using the sign is a useful way to indicate these two values.
