Solve each equation by square roots.
7. [tex]$x^2+4=29$[/tex]
Answer (2)
Subtract 4 from both sides: x 2 = 25 .
Take the square root of both sides: x = ± 25 .
Simplify the square root: x = ± 5 .
The solutions are x = 5 , − 5 .
Explanation
Problem Analysis Let's solve the equation x 2 + 4 = 29 using square roots. Our goal is to isolate x on one side of the equation.
Isolating the x^2 term First, we subtract 4 from both sides of the equation to isolate the x 2 term: x 2 + 4 − 4 = 29 − 4
Simplifying the equation This simplifies to: x 2 = 25
Taking the square root Now, we take the square root of both sides of the equation to solve for x :
x = ± 25
Finding the solutions Since the square root of 25 is 5, we have two possible solutions for x :
x = ± 5
Final Answer Therefore, the solutions are x = 5 and x = − 5 .
Examples
Understanding how to solve equations using square roots is useful in many areas, such as physics and engineering. For example, if you need to calculate the length of the side of a square given its area, you would use the square root. Suppose you have a square garden with an area of 25 square meters. To find the length of one side, you would solve the equation s 2 = 25 , where s is the side length. Taking the square root of both sides gives s = 5 meters. Thus, each side of the garden is 5 meters long.
To solve x 2 + 4 = 29 , we isolate x 2 by subtracting 4, resulting in x 2 = 25 . Taking the square root gives us the solutions x = 5 and x = − 5 . Thus, the answers are x = 5 , − 5 .
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