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Part 1

Solve the equation.

StartRoot 5 x plus 44 EndRoot5x+44minus−99equals=33xminus−55




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!

Solve the equation: StartRoot 5x + 44 EndRoot − 99 = 33x − 55

Learn how to solve an equation involving a square root and quadratic expression with a step-by-step breakdown. This problem covers algebraic concepts such as isolating square roots, squaring both sides, and applying the quadratic formula. Suitable for high school students in Grades 10-12.

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