Question

2.
√3x + 7 + √x + 1 =2​

Answer 1

x = -1

Explanation:

The usual approach to these is to square the radicals until they are gone.

`\displaystyle√(3x+7)+√(x+1)=2\\\\(3x+7) +2√((3x+7)(x+1))+(x+1) = 4\qquad\text{square both sides}\\\\2√((3x+7)(x+1))=-4x-4\qquad\text{subtract $4x+8$}\\\\(3x+7)(x+1)=(-2x-2)^2\qquad\text{divide by 2, square again}\\\\3x^2+10x +7=4x^2+8x+4\qquad\text{simplify}\\\\x^2-2x-3=0\qquad\text{subtract the left expression}\\\\(x-3)(x+1)=0\qquad\text{factor}\\\\x=3,\ x=-1\qquad\text{solutions to the quadratic}`

Each time the equation is squared, the possibility of an extraneous root is introduced. Here, x=3 is extraneous: it does not satisfy the original equation.

The solution is x = -1.

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Using a graphing calculator to solve the original equation can avoid extraneous solutions. The attachment shows only the solution x = -1. Rather than use f(x) = 2, we have rewritten the equation to f(x)-2 = 0. The graphing calculator is really good at showing the function values at the x-intercepts.