x Superscript 4 Baseline plus x squared minus 42equals0

The equation you've provided is:

\[
x^4 + x^2 - 42 = 0
\]

Let's solve this step by step.

### Step 1: Substitute \( y = x^2 \)

This substitution simplifies the equation. Let \( y = x^2 \), so the equation becomes:

\[
y^2 + y - 42 = 0
\]

This is a quadratic equation in terms of \( y \).

### Step 2: Solve the quadratic equation

We solve \( y^2 + y - 42 = 0 \) using the quadratic formula:

\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For this equation, \( a = 1 \), \( b = 1 \), and \( c = -42 \). Plugging in these values:

\[
y = \frac{-1 \pm \sqrt{1^2 - 4(1)(-42)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 168}}{2} = \frac{-1 \pm \sqrt{169}}{2}
\]

\[
y = \frac{-1 \pm 13}{2}
\]

Thus, we have two possible values for \( y \):

1. \( y = \frac{-1 + 13}{2} = \frac{12}{2} = 6 \)
2. \( y = \frac{-1 - 13}{2} = \frac{-14}{2} = -7 \)

### Step 3: Solve for \( x \)

Recall that \( y = x^2 \). Therefore:

1. If \( y = 6 \), then \( x^2 = 6 \), so \( x = \pm \sqrt{6} \).
2. If \( y = -7 \), then \( x^2 = -7 \), which has no real solution because a square cannot be negative in the real number system. Thus, there are no real solutions for \( y = -7 \).

### Final Answer

The real solutions to the equation \( x^4 + x^2 - 42 = 0 \) are:

\[
x = \pm \sqrt{6}
\]

Let me know if you need further clarification or additional details!

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Here are some related questions you could explore:

1. What happens if we solve this equation over the complex numbers?
2. How do we solve higher-degree polynomials in general?
3. What are some real-life applications of quartic equations?
4. Can this equation be solved graphically? If so, how?
5. What techniques can be used to solve similar equations without substitution?

**Tip:** Always check whether you need real or complex solutions when solving equations with squares!

Learn how to solve the quartic equation x^4 + x^2 - 42 = 0 step-by-step using quadratic substitution. This solution covers key algebraic techniques and formulas such as the quadratic formula, making it suitable for high school students (Grades 9-12).

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