https://file.aiquickdraw.com/mathbot/file/1726401950762wtoc1ipt.jpg

The equation shown in the image is:

\[
x^4 + 7x^2 = 8
\]

This is a **biquadratic equation** (as mentioned in the problem statement), which can be solved by substitution. Here's how we can solve it step by step:

### Step 1: Substitution
Let:
\[
y = x^2
\]
Then the equation becomes:
\[
y^2 + 7y = 8
\]

### Step 2: Rearrange the equation
Rearrange the equation to standard quadratic form:
\[
y^2 + 7y - 8 = 0
\]

### Step 3: Solve the quadratic equation
Now, solve the quadratic equation \(y^2 + 7y - 8 = 0\) using the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = 7\), and \(c = -8\).

\[
y = \frac{-7 \pm \sqrt{7^2 - 4(1)(-8)}}{2(1)}
\]
\[
y = \frac{-7 \pm \sqrt{49 + 32}}{2}
\]
\[
y = \frac{-7 \pm \sqrt{81}}{2}
\]
\[
y = \frac{-7 \pm 9}{2}
\]

So the two possible values for \(y\) are:
\[
y = \frac{-7 + 9}{2} = 1 \quad \text{or} \quad y = \frac{-7 - 9}{2} = -8
\]

### Step 4: Solve for \(x\)
Now, since \(y = x^2\), we substitute back:

- For \(y = 1\):
  \[
  x^2 = 1 \quad \Rightarrow \quad x = \pm 1
  \]

- For \(y = -8\):
  \[
  x^2 = -8 \quad \text{(no real solution since \(x^2\) cannot be negative in the real numbers)}
  \]

### Step 5: Final solution
The only real solutions are:
\[
x = 1 \quad \text{or} \quad x = -1
\]

Thus, the solution to the equation is:
\[
x = 1 \quad \text{or} \quad x = -1
\]

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

Here are 5 related questions to deepen your understanding:

1. What is a biquadratic equation, and how does it differ from a regular quadratic equation?
2. Can you explain why the substitution \(y = x^2\) simplifies the original equation?
3. What are the possible complex solutions for \(x^2 = -8\)?
4. How does the discriminant (\(b^2 - 4ac\)) help determine the nature of the solutions of a quadratic equation?
5. What methods can be used to solve quadratic equations other than the quadratic formula?

**Tip**: Whenever solving higher-degree polynomials, substitution techniques like this one can simplify the problem into a quadratic equation, making it much easier to solve!

x^4 + 7x^2 = 8 биквадрат тэгшитгэлийг бод.

Learn how to solve the biquadratic equation x^4 + 7x^2 = 8 with a step-by-step solution using the substitution method and the quadratic formula. This problem covers key algebraic concepts, including solving quadratic equations and higher-degree polynomials. Suitable for Grades 10-12 students.

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