The given equation is:

$\sqrt{5x + 44} - 99 = 33x - 55$

Step 1: Simplify both sides of the equation.

Start by isolating the square root term on one side. Add 99 to both sides:

$\sqrt{5x + 44} = 33x - 55 + 99$

Simplify the right side:

$\sqrt{5x + 44} = 33x + 44$

Step 2: Square both sides to eliminate the square root.

$(\sqrt{5x + 44})^2 = (33x + 44)^2$

This gives:

$5x + 44 = (33x + 44)^2$

Step 3: Expand the right side.

Now expand $(33x + 44)^2$:

$(33x + 44)^2 = 33x \times 33x + 2 \times 33x \times 44 + 44 \times 44$ $= 1089x^2 + 2904x + 1936$

Now the equation is:

$5x + 44 = 1089x^2 + 2904x + 1936$

Step 4: Move all terms to one side.

Subtract $5x + 44$ from both sides:

$0 = 1089x^2 + 2904x + 1936 - 5x - 44$

Simplify:

$0 = 1089x^2 + 2899x + 1892$

Step 5: Solve the quadratic equation.

Now we solve the quadratic equation:

$1089x^2 + 2899x + 1892 = 0$

We can apply the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Where:

• $a = 1089$
• $b = 2899$
• $c = 1892$

Now, let's compute the solution.The solutions to the quadratic equation are:

$x_1 \approx -1.146 \quad \text{and} \quad x_2 \approx -1.516$

Both are valid solutions as the discriminant ($162649$) is positive.

Would you like any further details or explanations?

Here are 5 related questions:

1. How do we handle square roots when solving equations?
2. What is the quadratic formula, and how does it work?
3. Can a quadratic equation have complex solutions?
4. What are the properties of discriminants in quadratic equations?
5. How can you check if your solution satisfies the original equation?

Tip: Always verify solutions by plugging them back into the original equation, especially after squaring both sides!