How to Solve Square Root Equation
Finding the square root of a number generally involves the use of a calculator, mainly because not all numbers have a whole number for a square root. Unfortunately, when looking for the square root of an equation, a calculator is generally not a source of help. While there are computer programs that can help, it isn't feasible to take such a program with you to class or to a testing session. As a result, it becomes necessary to learn how to solve a square root equation without relying on calculators and other technology for assistance.
- Difficulty:
- Moderate
Instructions
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1
Look at the problem. Knowing what the problem is asking for is the first step to finding a solution. For example, sqrt(4) is asking you to find the square root of 4; however, sqrt(x+1) = 4 is asking you to find x.
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2
Eliminate square roots by squaring both sides. In sqrt(x+1) = 4, you would change the equation to [sqrt(x+1)]^2 = 4^2. For sqrt(3 x + 1) = x-3, the problem would change to [sqrt(3x+1)]^2 = (x-3)^2.
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3
Simplify the equation. All of those square root symbols and powers of 2 make the problem look more confusing than it is. Take it one side at a time.
[sqrt(x+1)]^2 = 4^2 The power of 2 cancels the square root.
(x+1) = 4^2 4^2 is the same as 4 * 4.
(x+1) = 16 The equation in its simplest form.
[sqrt(3x+1)]^2 = (x-3)^2 Again, the power of 2 cancels the square root.
(3x+1)=(x-3)^2 Remember, (x-3)^2 is the same as (x-3)(x-3).
(3x+1)= x^2-6x+9 The equation in its simplest form, although it takes much longer to write.
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4
Solve for x.
(x+1)=16 The simple equation from Step three.
(x+1)-1=16-1 The first step to solving the equation is to move the x to one side, the numbers to the other. Do this by adding or subtracting from both sides.
x=15
(3x+1)=x^2-6x+9 Simple equation from Step three.
(3x+1)-(3x+1)=x^2-6x+9-(3x+1) In this case, both sides are an equation. To solve for x, the equation on the left must be moved as a whole.
0=x^2-6x+9-3x-1
0=x^2-9x+8 This is a quadratic equation, so the problem can be rewritten.
0=(x-8)(x-1)
0=(x-8) 0=(x-1) Set each part up as equal to 0, and then solve for x.
0+8=(x-8)+8 0+1=(x-1)+1
8=x 1=x There are two possible solutions to this problem.
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5
Check your work by substituting your solution for the variable in the original problem.
sqrt(x+1)=4
sqrt(15+1)= 4 Substitute, then solve.
sqrt16 = 4
4 = 4 Since the two sides match, the solution is correct.
sqrt(3x+1)= x-3 Remember, there were two solutions. Substitute one at a time, then solve.
sqrt(3*8+1) = 8-3
sqrt(24+1) = 5
sqrt(25) = 5
5 = 5 The two sides match, so 8 is a solution to the problem.
sqrt(3x+1) = x-3
sqrt(3*1+1) = 1-3
sqrt(3+1) = -2
sqrt(4) = -2
2 = -2 The two sides don't match. While 1 is an answer to the problem, it is not a solution. Instead, it is an answer created when the squares are eliminated.
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1
Tips & Warnings
Always check your work. It takes extra time, but it can help you see if you've made a mistake.
Even if an answer is not a solution, be sure to include it on your homework and exams. Some teachers and professors mark off points for incomplete answers if you do not include all possible solutions.
Always perform simplifying steps on both sides of the equation.
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Comments
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no comment!
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At last example, where it says "sqrt(4)!= -2" , it can be true. -2^2 = 2^2 because (-n)*(-n) becomes +n^2. That being said, the square root of 4 can be either +2 or -2, thus showing the equation as ±2 = -2, which IS true.
