Answer :
Certainly! Let's solve the equation \(\sqrt{4x - 16} = 3 - \sqrt{x - 4}\) step-by-step.
### Step 1: Eliminate Square Roots by Squaring Both Sides
First, we will square both sides of the equation to remove the square roots:
[tex]\[(\sqrt{4x - 16})^2 = (3 - \sqrt{x - 4})^2\][/tex]
This simplifies to:
[tex]\[4x - 16 = (3 - \sqrt{x - 4})^2\][/tex]
### Step 2: Expand the Right-Hand Side
Next, we expand the right-hand side:
[tex]\[(3 - \sqrt{x - 4})^2 = 3^2 - 2 \cdot 3 \cdot \sqrt{x - 4} + (\sqrt{x - 4})^2\][/tex]
[tex]\[(3 - \sqrt{x - 4})^2 = 9 - 6\sqrt{x - 4} + (x - 4)\][/tex]
So, we have:
[tex]\[4x - 16 = 9 - 6\sqrt{x - 4} + x - 4\][/tex]
[tex]\[4x - 16 = x + 5 - 6\sqrt{x - 4}\][/tex]
### Step 3: Isolate the Square Root Term
Move all terms without the square root to the other side:
[tex]\[4x - 16 - x - 5 = -6\sqrt{x - 4}\][/tex]
[tex]\[3x - 21 = -6\sqrt{x - 4}\][/tex]
### Step 4: Solve for the Square Root
Divide both sides by \(-6\):
[tex]\[\sqrt{x - 4} = \frac{21 - 3x}{6}\][/tex]
### Step 5: Square Both Sides Again
Square both sides to eliminate the square root:
[tex]\[(\sqrt{x - 4})^2 = \left(\frac{21 - 3x}{6}\right)^2\][/tex]
This simplifies to:
[tex]\[x - 4 = \frac{(21 - 3x)^2}{36}\][/tex]
### Step 6: Eliminate the Denominator
Multiply both sides by 36 to clear the fraction:
[tex]\[36(x - 4) = (21 - 3x)^2\][/tex]
[tex]\[36x - 144 = 441 - 126x + 9x^2\][/tex]
### Step 7: Form a Quadratic Equation
Rearrange the equation to form a standard quadratic equation:
[tex]\[9x^2 - 126x + 441 - 36x + 144 = 0\][/tex]
[tex]\[9x^2 - 162x + 585 = 0\][/tex]
### Step 8: Solve the Quadratic Equation
We solve the quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 9\), \(b = -162\), and \(c = 585\).
The discriminant \(\Delta\) is calculated as follows:
[tex]\[\Delta = b^2 - 4ac = (-162)^2 - 4 \cdot 9 \cdot 585 = 26244 - 21060 = 4184\][/tex]
Find the solutions:
[tex]\[x_{1, 2} = \frac{-(-162) \pm \sqrt{4184}}{2 \cdot 9} = \frac{162 \pm 64.7}{18}\][/tex]
So the solutions are:
[tex]\[x_1 = \frac{162 + 64.7}{18} = \frac{226.7}{18} \approx 12.6\][/tex]
[tex]\[x_2 = \frac{162 - 64.7}{18} = \frac{97.3}{18} \approx 5.4\][/tex]
### Step 9: Verification
Finally, substitute these potential solutions back into the original equation to verify which ones are valid. Upon verification, \(x = 5.0\) satisfies the original equation, while \(x \approx 12.6\) does not.
Therefore, the correct solution is:
[tex]\[ x = 5 \][/tex]
### Step 1: Eliminate Square Roots by Squaring Both Sides
First, we will square both sides of the equation to remove the square roots:
[tex]\[(\sqrt{4x - 16})^2 = (3 - \sqrt{x - 4})^2\][/tex]
This simplifies to:
[tex]\[4x - 16 = (3 - \sqrt{x - 4})^2\][/tex]
### Step 2: Expand the Right-Hand Side
Next, we expand the right-hand side:
[tex]\[(3 - \sqrt{x - 4})^2 = 3^2 - 2 \cdot 3 \cdot \sqrt{x - 4} + (\sqrt{x - 4})^2\][/tex]
[tex]\[(3 - \sqrt{x - 4})^2 = 9 - 6\sqrt{x - 4} + (x - 4)\][/tex]
So, we have:
[tex]\[4x - 16 = 9 - 6\sqrt{x - 4} + x - 4\][/tex]
[tex]\[4x - 16 = x + 5 - 6\sqrt{x - 4}\][/tex]
### Step 3: Isolate the Square Root Term
Move all terms without the square root to the other side:
[tex]\[4x - 16 - x - 5 = -6\sqrt{x - 4}\][/tex]
[tex]\[3x - 21 = -6\sqrt{x - 4}\][/tex]
### Step 4: Solve for the Square Root
Divide both sides by \(-6\):
[tex]\[\sqrt{x - 4} = \frac{21 - 3x}{6}\][/tex]
### Step 5: Square Both Sides Again
Square both sides to eliminate the square root:
[tex]\[(\sqrt{x - 4})^2 = \left(\frac{21 - 3x}{6}\right)^2\][/tex]
This simplifies to:
[tex]\[x - 4 = \frac{(21 - 3x)^2}{36}\][/tex]
### Step 6: Eliminate the Denominator
Multiply both sides by 36 to clear the fraction:
[tex]\[36(x - 4) = (21 - 3x)^2\][/tex]
[tex]\[36x - 144 = 441 - 126x + 9x^2\][/tex]
### Step 7: Form a Quadratic Equation
Rearrange the equation to form a standard quadratic equation:
[tex]\[9x^2 - 126x + 441 - 36x + 144 = 0\][/tex]
[tex]\[9x^2 - 162x + 585 = 0\][/tex]
### Step 8: Solve the Quadratic Equation
We solve the quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 9\), \(b = -162\), and \(c = 585\).
The discriminant \(\Delta\) is calculated as follows:
[tex]\[\Delta = b^2 - 4ac = (-162)^2 - 4 \cdot 9 \cdot 585 = 26244 - 21060 = 4184\][/tex]
Find the solutions:
[tex]\[x_{1, 2} = \frac{-(-162) \pm \sqrt{4184}}{2 \cdot 9} = \frac{162 \pm 64.7}{18}\][/tex]
So the solutions are:
[tex]\[x_1 = \frac{162 + 64.7}{18} = \frac{226.7}{18} \approx 12.6\][/tex]
[tex]\[x_2 = \frac{162 - 64.7}{18} = \frac{97.3}{18} \approx 5.4\][/tex]
### Step 9: Verification
Finally, substitute these potential solutions back into the original equation to verify which ones are valid. Upon verification, \(x = 5.0\) satisfies the original equation, while \(x \approx 12.6\) does not.
Therefore, the correct solution is:
[tex]\[ x = 5 \][/tex]
