x4+29x2−29=0
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Part 1
What is the solution​ set? Select the correct choice below and fill in any answer boxes within your choice.
A.
StartSet nothing EndSetenter your response here
​(Use a comma to separate answers as needed. Round to the nearest hundredth as​ needed.)
B.
There are no solutions.

Let's solve the equation \( x^4 + 29x^2 - 29 = 0 \) to determine the solution set.

### Step 1: Substitute \( y = x^2 \)
This will simplify the quartic equation to a quadratic form:
\[
y^2 + 29y - 29 = 0
\]

### Step 2: Solve the quadratic equation
We can solve for \( y \) using the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 1 \), \( b = 29 \), and \( c = -29 \).

Substituting these values:
\[
y = \frac{-29 \pm \sqrt{29^2 - 4(1)(-29)}}{2(1)}
\]
\[
y = \frac{-29 \pm \sqrt{841 + 116}}{2}
\]
\[
y = \frac{-29 \pm \sqrt{957}}{2}
\]

Now, calculate the discriminant:
\[
\sqrt{957} \approx 30.93
\]
So:
\[
y_1 = \frac{-29 + 30.93}{2} \approx \frac{1.93}{2} \approx 0.97
\]
\[
y_2 = \frac{-29 - 30.93}{2} \approx \frac{-59.93}{2} \approx -29.97
\]

### Step 3: Find \( x \) from \( y = x^2 \)
1. For \( y_1 = 0.97 \):
   \[
   x^2 = 0.97 \implies x = \pm \sqrt{0.97} \approx \pm 0.99
   \]

2. For \( y_2 = -29.97 \):
   Since \( y_2 \) is negative and \( x^2 \) cannot be negative for real numbers, this value does not yield any real solutions.

### Conclusion:
The only real solutions are \( x \approx 0.99 \) and \( x \approx -0.99 \). 

Thus, the solution set is:
\[
\text{Option A: } x \approx \pm 0.99
\]

Would you like more details on any step, or do you have any questions?

---

**5 Relative Questions:**
1. How do you solve a quadratic equation using the quadratic formula?
2. What is the importance of checking the discriminant in quadratic equations?
3. How would the solution change if the discriminant were negative?
4. Can a quartic equation always be reduced to a quadratic equation?
5. How do you interpret the solutions of \( x^2 = -1 \) in the context of complex numbers?

**Tip:** Always verify the discriminant first to check if the quadratic equation has real solutions before proceeding to solve.

x^4 + 29x^2 - 29 = 0. What is the solution set? Select the correct choice below and fill in any answer boxes within your choice.

Learn how to solve the quartic equation x^4 + 29x^2 - 29 by reducing it to a quadratic equation. This detailed solution uses the quadratic formula and explores discriminant calculation. Ideal for students in Grades 9-11.

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