Instructions:

Solve the following equations and find the real roots. Provide your final answer in the space provided.

Questions:

1) Find the real roots of the equation:

• \( \frac{9}{x^4} + \frac{8}{x^2} = 1 \)

Answer: _______________________

2) Solve:

• \( x^4 - 13x^2 + 36 = 0 \)

Answer: _______________________

3) Find the real roots of the equation:

• \( x^2 - 2 = \frac{8}{x^2} \)

Answer: _______________________

4) Solve:

• \( x^6 - 7x^3 - 8 = 0 \)

Answer: _______________________

5) Solve:

• \( 2x^4 - 11x^2 + 5 = 0 \)

Answer: _______________________

6) Solve the equation:

• \( \frac{9}{x^4} + \frac{5}{x^2} = 4 \)

Answer: _______________________

7) Solve:

• \( 2x - 9\sqrt{x} + 10 = 0 \)

Answer: _______________________

8) Solve:

• \( \sqrt{x}(\sqrt{x} + 1) = 6 \)

Answer: _______________________

9) Find the x-coordinates of points A and B where the curve \( y = 2\sqrt{x} \) and the line \( 3y = x + 8 \) intersect.

Answer: _______________________

10) Find the values of \( a \), \( b \), and \( c \) for the quadratic equation \( y = ax^2 + bx + c \), given that the graph crosses the x-axis at \( (1, 0) \) and \( \left(\frac{49}{4}, 0\right) \) and meets the y-axis at \( (0, 7) \).

Answer: a = __________, b = __________, c = __________

Answer Key:

• 1) \( x = 3 \) or \( x = -3 \)
• 2) \( x = \pm 3 \) or \( x = \pm 2 \)
• 3) \( x = 2 \) or \( x = -2 \)
• 4) \( x = 2 \) or \( x = -1 \)
• 5) \( x = \pm \sqrt{5} \) or \( x = \pm \frac{\sqrt{2}}{2} \)
• 6) \( x = \frac{3}{2} \) or \( x = -\frac{3}{2} \)
• 7) \( x = 6.25 \) or \( x = 4 \)
• 8) \( x = 4 \)
• 9) \( x = 16 \) and \( x = 4 \)
• 10) \( a = \frac{4}{7}, b = -\frac{53}{7}, c = 7 \)