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college algebra / benchmark 2: checkpoints 4-5 / benchmark 2: checkpoints 4-5
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solve the radical equation \\(\sqrt{2x + 3} - \sqrt{x + 2} = 2\\).
select one:
a. no solution
b. \\(x = -23, 1\\)
c. \\(x = 23, -1\\)
d. \\(x = 23\\)

Explanation:

Step1: Isolate one radical

We start with the equation $\sqrt{2x + 3}-\sqrt{x + 2}=2$. Let's isolate $\sqrt{2x + 3}$ by adding $\sqrt{x + 2}$ to both sides:
$\sqrt{2x + 3}=2+\sqrt{x + 2}$

Step2: Square both sides

Now, square both sides of the equation to eliminate the square root on the left. Recall that $(a + b)^2=a^2+2ab + b^2$. So, squaring the left side gives $2x + 3$, and squaring the right side:
$(2+\sqrt{x + 2})^2=2^2+2\times2\times\sqrt{x + 2}+(\sqrt{x + 2})^2=4 + 4\sqrt{x + 2}+x + 2=x + 6+4\sqrt{x + 2}$
So we have the equation:
$2x + 3=x + 6+4\sqrt{x + 2}$

Step3: Simplify the equation

Subtract $x$ and $6$ from both sides to isolate the term with the square root:
$2x+3 - x - 6=4\sqrt{x + 2}$
$x - 3=4\sqrt{x + 2}$

Step4: Square both sides again

Square both sides to eliminate the square root. The left side is $(x - 3)^2=x^2-6x + 9$, and the right side is $(4\sqrt{x + 2})^2=16(x + 2)=16x+32$
So we get the quadratic equation:
$x^2-6x + 9=16x+32$

Step5: Solve the quadratic equation

Bring all terms to one side:
$x^2-6x + 9-16x - 32=0$
$x^2-22x - 23=0$
Factor the quadratic equation. We need two numbers that multiply to $- 23$ and add to $-22$. The numbers are $-23$ and $1$:
$(x - 23)(x + 1)=0$
Setting each factor equal to zero gives $x=23$ or $x=-1$

Step6: Check for extraneous solutions

We need to check these solutions in the original equation because squaring both sides can introduce extraneous solutions.

Check $x = 23$:

Left side of original equation: $\sqrt{2\times23 + 3}-\sqrt{23+2}=\sqrt{46 + 3}-\sqrt{25}=\sqrt{49}-5 = 7 - 5=2$
Right side: $2$
So $x = 23$ is a valid solution.

Check $x=-1$:

Left side of original equation: $\sqrt{2\times(-1)+3}-\sqrt{-1 + 2}=\sqrt{-2 + 3}-\sqrt{1}=\sqrt{1}-1=1 - 1 = 0$
Right side: $2$
Since $0
eq2$, $x=-1$ is an extraneous solution.

Answer:

D. $x = 23$