Things You'll Need:
- Pencil
- Paper
- Eraser
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Step 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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Step 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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Step 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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Step 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-10=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)+18=x 1=x There are two possible solutions to this problem.
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Step 5
Check your work by substituting your solution for the variable in the original problem.sqrt(x+1)=4sqrt(15+1)= 4 Substitute, then solve.sqrt16 = 44 = 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) = 8-3sqrt(24+1) = 5sqrt(25) = 55 = 5 The two sides match, so 8 is a solution to the problem.sqrt(3x+1) = x-3sqrt(3*1+1) = 1-3sqrt(3+1) = -2sqrt(4) = -22 = -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.
