Let's solve the equation:

Given:

$X = 21\sqrt{X} + 130$

Step 1: Isolate the square root term.

Subtract $130$ from both sides: $X - 130 = 21\sqrt{X}$

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

$(X - 130)^2 = (21\sqrt{X})^2$ $(X - 130)^2 = 441X$

Step 3: Expand $(X - 130)^2$.

$X^2 - 260X + 16900 = 441X$

Step 4: Rearrange into standard quadratic form.

$X^2 - 260X - 441X + 16900 = 0$ $X^2 - 701X + 16900 = 0$

Step 5: Solve the quadratic equation.

The quadratic equation is: $X^2 - 701X + 16900 = 0$ Use the quadratic formula: $X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here:

• $a = 1$,
• $b = -701$,
• $c = 16900$.

Substitute these values: $X = \frac{-(-701) \pm \sqrt{(-701)^2 - 4(1)(16900)}}{2(1)}$ $X = \frac{701 \pm \sqrt{491401 - 67600}}{2}$ $X = \frac{701 \pm \sqrt{423801}}{2}$ $X = \frac{701 \pm 651.61}{2}$

Step 6: Calculate the two possible solutions.

1. $X = \frac{701 + 651.61}{2} = \frac{1352.61}{2} = 676.305$
2. $X = \frac{701 - 651.61}{2} = \frac{49.39}{2} = 24.695$

Step 7: Verify both solutions.

Plug each back into the original equation to check.

• For $X = 676.305$: $676.305 = 21\sqrt{676.305} + 130$ $676.305 \approx 21(26) + 130$ $676.305 \approx 676$ Valid solution.

• For $X = 24.695$: $24.695 = 21\sqrt{24.695} + 130$ $24.695 \approx 21(5) + 130$ $24.695 \neq 235$ Invalid solution.

Final Answer:

$X = 676.305$

Would you like further clarification or details?
Here are some related questions for exploration:

1. How does squaring both sides impact equations with square roots?
2. Can the quadratic formula be used for equations with higher powers?
3. Why is verifying solutions important in equations with square roots?
4. What is the geometric interpretation of this equation?
5. How do rounding errors affect the precision of solutions?

Tip: Always verify solutions when working with equations that involve squaring, as extraneous solutions may arise.