Quadratic Equations and Functions
Solve Quadratic Equations Using the Square Root Property
Learning Objectives
By the end of this section, you will be able to:
- Solve quadratic equations of the form using the Square Root Property
- Solve quadratic equations of the form using the Square Root Property
Before you get started, take this readiness quiz.
A quadratic equation is an equation of the form ax2 + bx + c = 0, where . Quadratic equations differ from linear equations by including a quadratic term with the variable raised to the second power of the form ax2. We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.
We have seen that some quadratic equations can be solved by factoring. In this chapter, we will learn three other methods to use in case a quadratic equation cannot be factored.
Solve Quadratic Equations of the form using the Square Root Property
We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation x2 = 9.
We can easily use factoring to find the solutions of similar equations, like x2 = 16 and x2 = 25, because 16 and 25 are perfect squares. In each case, we would get two solutions, and
But what happens when we have an equation like x2 = 7? Since 7 is not a perfect square, we cannot solve the equation by factoring.
Previously we learned that since 169 is the square of 13, we can also say that 13 is a square root of 169. Also, (−13)2 = 169, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169. So, every positive number has two square roots—one positive and one negative. We earlier defined the square root of a number in this way:
Since these equations are all of the form x2 = k, the square root definition tells us the solutions are the two square roots of k. This leads to the Square Root Property.
If x2 = k, then
Notice that the Square Root Property gives two solutions to an equation of the form x2 = k, the principal square root of and its opposite. We could also write the solution as We read this as x equals positive or negative the square root of k.
Now we will solve the equation x2 = 9 again, this time using the Square Root Property.
What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation x2 = 7.
We cannot simplify , so we leave the answer as a radical.
Solve:
Solve:
Solve:
The steps to take to use the Square Root Property to solve a quadratic equation are listed here.
- Isolate the quadratic term and make its coefficient one.
- Use Square Root Property.
- Simplify the radical.
- Check the solutions.
In order to use the Square Root Property, the coefficient of the variable term must equal one. In the next example, we must divide both sides of the equation by the coefficient 3 before using the Square Root Property.
Solve:
| The quadratic term is isolated.