To solve the equation and verify, follow these steps:
Step 1: Start by isolating the square root term on one side of the equation.
\sqrt{3x+7} + 1 = x
Subtract 1 from both sides:
\sqrt{3x+7} = x - 1
Step 2: Square both sides of the equation to get rid of the square root.
(\sqrt{3x+7})^2 = (x - 1)^2
3x + 7 = x^2 - 2x + 1
Step 3: Rearrange the equation to get a quadratic equation.
x^2 - 2x + 1 - 3x - 7 = 0
x^2 - 5x - 6 = 0
Step 4: Solve the quadratic equation by factoring or using the quadratic formula.
(x - 6)(x + 1) = 0
Setting each factor equal to zero:
x - 6 = 0 or x + 1 = 0
Solving for x:
x = 6 or x = -1
Step 5: Verify the solutions by substituting them back into the original equation.
For x = 6:
\sqrt{3(6)+7} + 1 = 6
\sqrt{18+7} + 1 = 6
\sqrt{25} + 1 = 6
5 + 1 = 6
6 = 6
The equation is verified for x = 6.
For x = -1:
\sqrt{3(-1)+7} + 1 = -1
\sqrt{3-1+7} + 1 = -1
\sqrt{9} + 1 = -1
3 + 1 = -1
4 = -1
The equation is not verified for x = -1.
Therefore, the only solution that satisfies the original equation is x = 6.
solve the equation and varify \sqrt(3x+7)+1=x
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