Find all real and imaginary solutions to the equation. Check your answer.

Question

Find all real and imaginary solutions to the equation. Check your answer. $$ x^{4}-16 x^{2}+63=0 $$

UpStudy AI · Accepted Answer

To find all real and imaginary solutions to the equation:

$$
x^{4} - 16x^{2} + 63 = 0
$$

we can solve it step-by-step as follows:

1. **Recognize the Equation as a Biquadratic:**
   
   The given equation is a biquadratic equation because it contains even powers of $$ x $$. We can simplify it by making a substitution.

2. **Substitute $$ y = x^2 $$:**
   
   Let $$ y = x^2 $$. The equation becomes:

$$
   y^2 - 16y + 63 = 0
   $$

3. **Solve the Quadratic Equation for $$ y $$:**

To solve $$ y^2 - 16y + 63 = 0 $$, we can use the quadratic formula:

$$
   y = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot 63}}{2} = \frac{16 \pm \sqrt{256 - 252}}{2} = \frac{16 \pm \sqrt{4}}{2} = \frac{16 \pm 2}{2}
   $$

This gives us two solutions for $$ y $$:

$$
   y_1 = \frac{16 + 2}{2} = 9 \quad \text{and} \quad y_2 = \frac{16 - 2}{2} = 7
   $$

4. **Back-Substitute to Find $$ x $$:**

Since $$ y = x^2 $$, we now have:

$$
   x^2 = 9 \quad \Rightarrow \quad x = \pm 3
   $$
   
   $$
   x^2 = 7 \quad \Rightarrow \quad x = \pm \sqrt{7}
   $$

5. **List All Solutions:**

The equation has four real solutions:

$$
   x = 3, \quad x = -3, \quad x = \sqrt{7}, \quad x = -\sqrt{7}
   $$

6. **Verify the Solutions:**

Substitute each solution back into the original equation to verify:

- For $$ x = 3 $$:
     $$
     3^4 - 16 \cdot 3^2 + 63 = 81 - 144 + 63 = 0
     $$
   
   - For $$ x = -3 $$:
     $$
     (-3)^4 - 16 \cdot (-3)^2 + 63 = 81 - 144 + 63 = 0
     $$
   
   - For $$ x = \sqrt{7} $$:
     $$
     (\sqrt{7})^4 - 16 \cdot (\sqrt{7})^2 + 63 = 49 - 112 + 63 = 0
     $$
   
   - For $$ x = -\sqrt{7} $$:
     $$
     (-\sqrt{7})^4 - 16 \cdot (-\sqrt{7})^2 + 63 = 49 - 112 + 63 = 0
     $$
   
   All solutions satisfy the original equation.

**Final Answer:**

All four roots are real numbers  3, –3, √7, and –√7. Thus, the solutions are

x = ± 3 and x = ± √7.
The solutions are $$ x = 3, -3, \sqrt{7}, -\sqrt{7} $$.

Find all real and imaginary solutions to the equation. Check your answer.

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