To solve the equation [tex]\(\sqrt{4x - 3} - \sqrt{x - 3} = 3\)[/tex], we'll proceed step-by-step.
1. Isolate one of the square root terms:
[tex]\[\sqrt{4x - 3} = \sqrt{x - 3} + 3\][/tex]
2. Square both sides of the equation to remove the square root on one side:
[tex]\[(\sqrt{4x - 3})^2 = (\sqrt{x - 3} + 3)^2\][/tex]
3. This simplifies to:
[tex]\[4x - 3 = (x - 3) + 6\sqrt{x - 3} + 9\][/tex]
[tex]\[4x - 3 = x - 3 + 6\sqrt{x - 3} + 9\][/tex]
[tex]\[4x - 3 = x + 6\sqrt{x - 3} + 6\][/tex]
4. Move all terms to one side to isolate the square root term:
[tex]\[4x - 3 - x - 6 = 6\sqrt{x - 3}\][/tex]
[tex]\[3x - 9 = 6\sqrt{x - 3}\][/tex]
5. Simplify:
[tex]\[3(x - 3) = 6\sqrt{x - 3}\][/tex]
[tex]\[x - 3 = 2\sqrt{x - 3}\][/tex]
6. Square both sides again to remove the remaining square root:
[tex]\[(x - 3)^2 = (2\sqrt{x - 3})^2\][/tex]
[tex]\[x^2 - 6x + 9 = 4(x - 3)\][/tex]
7. Expand and simplify:
[tex]\[x^2 - 6x + 9 = 4x - 12\][/tex]
[tex]\[x^2 - 6x + 9 - 4x + 12 = 0\][/tex]
[tex]\[x^2 - 10x + 21 = 0\][/tex]
8. Solve the quadratic equation using the quadratic formula [tex]\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = -10\)[/tex], and [tex]\(c = 21\)[/tex]:
[tex]\[x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 21}}{2 \cdot 1}\][/tex]
[tex]\[x = \frac{10 \pm \sqrt{100 - 84}}{2}\][/tex]
[tex]\[x = \frac{10 \pm \sqrt{16}}{2}\][/tex]
[tex]\[x = \frac{10 \pm 4}{2}\][/tex]
[tex]\[x = \frac{10 + 4}{2} \quad \text{or} \quad x = \frac{10 - 4}{2}\][/tex]
[tex]\[x = 7 \quad \text{or} \quad x = 3\][/tex]
9. Verify the solutions by substituting [tex]\(x = 3\)[/tex] and [tex]\(x = 7\)[/tex] back into the original equation:
- For [tex]\(x = 3\)[/tex]:
[tex]\[\sqrt{4(3) - 3} - \sqrt{3 - 3} = 3\][/tex]
[tex]\[\sqrt{12 - 3} - \sqrt{0} = 3\][/tex]
[tex]\[\sqrt{9} - 0 = 3\][/tex]
[tex]\[3 = 3\][/tex] (True)
- For [tex]\(x = 7\)[/tex]:
[tex]\[\sqrt{4(7) - 3} - \sqrt{7 - 3} = 3\][/tex]
[tex]\[\sqrt{28 - 3} - \sqrt{4} = 3\][/tex]
[tex]\[\sqrt{25} - 2 = 3\][/tex]
[tex]\[5 - 2 = 3\][/tex] (True)
Both solutions are valid; therefore, the solution to the equation [tex]\(\sqrt{4x - 3} - \sqrt{x - 3} = 3\)[/tex] is:
[tex]\[\boxed{3, 7}\][/tex]
