Recall that a quadratic equation is in standard form if it is equal to 0:
where a, b, and c are real numbers and . A solution to such an equation is called a root. Quadratic equations can have two real solutions, one real solution, or no real solution. If the quadratic expression on the left factors, then we can solve it by factoring. A review of the steps used to solve by factoring follow:
Step 1: Express the quadratic equation in standard form.
Step 2: Factor the quadratic expression.
Step 3: Apply the zero-product property and set each variable factor equal to 0.
Step 4: Solve the resulting linear equations.
For example, we can solve by factoring as follows:
The two solutions are −2 and 2. The goal in this section is to develop an alternative method that can be used to easily solve equations where b = 0, giving the form
The equation is in this form and can be solved by first isolating .
If we take the square root of both sides of this equation, we obtain the following:
Here we see that and are solutions to the resulting equation. In general, this describes the square root property; for any real number k,
The notation “±” is read “plus or minus” and is used as compact notation that indicates two solutions. Hence the statement indicates that or . Applying the square root property as a means of solving a quadratic equation is called extracting the roots.
Example 1: Solve: .
Solution: Begin by isolating the square.
Next, apply the square root property.
Answer: The solutions are −5 and 5. The check is left to the reader.
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Answer:
Extracting Square Roots
Recall that a quadratic equation is in standard form if it is equal to 0:
where a, b, and c are real numbers and . A solution to such an equation is called a root. Quadratic equations can have two real solutions, one real solution, or no real solution. If the quadratic expression on the left factors, then we can solve it by factoring. A review of the steps used to solve by factoring follow:
Step 1: Express the quadratic equation in standard form.
Step 2: Factor the quadratic expression.
Step 3: Apply the zero-product property and set each variable factor equal to 0.
Step 4: Solve the resulting linear equations.
For example, we can solve by factoring as follows:
The two solutions are −2 and 2. The goal in this section is to develop an alternative method that can be used to easily solve equations where b = 0, giving the form
The equation is in this form and can be solved by first isolating .
If we take the square root of both sides of this equation, we obtain the following:
Here we see that and are solutions to the resulting equation. In general, this describes the square root property; for any real number k,
The notation “±” is read “plus or minus” and is used as compact notation that indicates two solutions. Hence the statement indicates that or . Applying the square root property as a means of solving a quadratic equation is called extracting the roots.
Example 1: Solve: .
Solution: Begin by isolating the square.
Next, apply the square root property.
Answer: The solutions are −5 and 5. The check is left to the reader.