Solve by factoring or finding square roots.
[tex]x^2-49=0[/tex]
Answer (1)
Rewrite the equation as x 2 = 49 .
Take the square root of both sides: x = ± 49 .
Simplify to find the two possible values of x : x = 7 or x = − 7 .
The solutions to the equation x 2 − 49 = 0 are 7 , − 7 .
Explanation
Understanding the Problem We are given the equation x 2 − 49 = 0 . Our goal is to solve for x , which means finding all values of x that make the equation true. We can solve this equation by factoring or by using square roots.
Isolating x squared Let's solve by finding square roots. First, we isolate x 2 by adding 49 to both sides of the equation: x 2 − 49 + 49 = 0 + 49 x 2 = 49
Taking the Square Root Now, we take the square root of both sides of the equation. Remember that when we take the square root of a number, we must consider both the positive and negative square roots: x = ± 49 Since 49 = 7 , we have: x = ± 7
Solving by Factoring This gives us two solutions: x = 7 and x = − 7 . Alternatively, we can solve by factoring. We recognize that x 2 − 49 is a difference of squares, which can be factored as ( x − 7 ) ( x + 7 ) . So, we have: ( x − 7 ) ( x + 7 ) = 0
Setting Factors to Zero To find the solutions, we set each factor equal to zero: x − 7 = 0 or x + 7 = 0
Finding the Solutions Solving these equations, we get: x = 7 or x = − 7 Thus, the solutions are x = 7 and x = − 7 .
Final Answer Therefore, the solutions to the equation x 2 − 49 = 0 are x = 7 and x = − 7 .
Examples
Imagine you are designing a square garden with an area of 49 square feet. This problem helps you determine the possible side lengths of the garden. Since area of square is x 2 , where x is the side length, the equation becomes x 2 = 49 . Solving this equation gives you the possible side lengths, which are 7 feet (we only consider the positive root since length cannot be negative). This concept is useful in various scenarios, such as calculating areas, designing layouts, and solving geometric problems.
