Certainly! Let's go through the solution step-by-step:
We are trying to solve the equation:
[tex]\[
\sqrt{4x - 3} - \sqrt{x - 3} = 3
\][/tex]
Step 1: Isolate one of the square roots
First, we need to isolate one of the square roots. Let's isolate [tex]\(\sqrt{4x - 3}\)[/tex]:
[tex]\[
\sqrt{4x - 3} = \sqrt{x - 3} + 3
\][/tex]
Step 2: Square both sides to eliminate the square root
Next, we square both sides of the equation in order to eliminate the square root:
[tex]\[
(\sqrt{4x - 3})^2 = (\sqrt{x - 3} + 3)^2
\][/tex]
This gives us:
[tex]\[
4x - 3 = (x - 3) + 6\sqrt{x - 3} + 9
\][/tex]
Simplify the right-hand side of the equation:
[tex]\[
4x - 3 = x - 3 + 6\sqrt{x - 3} + 9
\][/tex]
Combine like terms:
[tex]\[
4x - 3 = x + 6\sqrt{x - 3} + 6
\][/tex]
Step 3: Isolate the term with the square root
Next, we need to isolate the term with the square root:
[tex]\[
4x - x - 3 - 6 = 6\sqrt{x - 3}
\][/tex]
Simplify the equation:
[tex]\[
3x - 9 = 6\sqrt{x - 3}
\][/tex]
Step 4: Square both sides again
Square both sides of the equation again to eliminate the remaining square root:
[tex]\[
(3x - 9)^2 = (6\sqrt{x - 3})^2
\][/tex]
This gives us:
[tex]\[
(3x - 9)^2 = 36(x - 3)
\][/tex]
Expand the left-hand side:
[tex]\[
9x^2 - 54x + 81 = 36x - 108
\][/tex]
Step 5: Bring all terms to one side
Move all terms to one side to set the equation to zero:
[tex]\[
9x^2 - 54x + 81 - 36x + 108 = 0
\][/tex]
Combine like terms:
[tex]\[
9x^2 - 90x + 189 = 0
\][/tex]
Step 6: Simplify the quadratic equation
Divide the entire equation by 9 to simplify it:
[tex]\[
x^2 - 10x + 21 = 0
\][/tex]
Step 7: Solve the quadratic equation
Now we need to solve this quadratic equation. We can factor it:
[tex]\[
(x - 3)(x - 7) = 0
\][/tex]
So, the solutions to the equation are:
[tex]\[
x = 3 \quad \text{and} \quad x = 7
\][/tex]
Step 8: Verify the solutions
We need to check whether both solutions satisfy the original equation.
For [tex]\(x = 3\)[/tex]:
[tex]\[
\sqrt{4(3) - 3} - \sqrt{3 - 3} = \sqrt{12 - 3} - \sqrt{0} = \sqrt{9} - 0 = 3 \quad \text{(which satisfies the equation)}
\][/tex]
For [tex]\(x = 7\)[/tex]:
[tex]\[
\sqrt{4(7) - 3} - \sqrt{7 - 3} = \sqrt{28 - 3} - \sqrt{4} = \sqrt{25} - 2 = 5 - 2 = 3 \quad \text{(which also satisfies the equation)}
\][/tex]
Therefore, the solutions to the equation are:
3 and 7
