$$ \quad \text { Solve the equation. } $$ $$ x^{4}-4 x^{2}-60=0 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
x^{4} - 4x^{2} - 60 = 0
$$

we can use substitution and factorization techniques. Here's a step-by-step solution:

### Step 1: Let’s Make a Substitution
Let $$ y = x^{2} $$. This substitution simplifies the equation by reducing its degree.

$$
y^{2} - 4y - 60 = 0
$$

### Step 2: Solve the Quadratic Equation in $$ y $$
Now, solve the quadratic equation for $$ y $$:

$$
y^{2} - 4y - 60 = 0
$$

Using the quadratic formula $$ y = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} $$, where $$ a = 1 $$, $$ b = -4 $$, and $$ c = -60 $$:

$$
y = \frac{-(-4) \pm \sqrt{(-4)^{2} - 4(1)(-60)}}{2(1)} = \frac{4 \pm \sqrt{16 + 240}}{2} = \frac{4 \pm \sqrt{256}}{2}
$$

$$
y = \frac{4 \pm 16}{2}
$$

This gives two solutions for $$ y $$:

$$
y = \frac{4 + 16}{2} = 10 \quad \text{and} \quad y = \frac{4 - 16}{2} = -6
$$

### Step 3: Back-Substitute $$ y = x^{2} $$

1. **First Case:** $$ y = 10 $$

$$
x^{2} = 10 \implies x = \pm \sqrt{10}
$$

2. **Second Case:** $$ y = -6 $$

$$
x^{2} = -6 \implies x = \pm \sqrt{-6} = \pm i\sqrt{6}
$$

### Step 4: Combine All Solutions

The solutions to the original equation $$ x^{4} - 4x^{2} - 60 = 0 $$ are:

$$
x = \sqrt{10}, \quad x = -\sqrt{10}, \quad x = i\sqrt{6}, \quad x = -i\sqrt{6}
$$

### Final Answer

$$
x = \pm \sqrt{10} \quad \text{and} \quad x = \pm i\,\sqrt{6}
$$
The solutions are $$ x = \sqrt{10} $$, $$ x = -\sqrt{10} $$, $$ x = i\sqrt{6} $$, and $$ x = -i\sqrt{6} $$.

\[ \quad \text { Solve the equation. } \] \[ x^{4}-4 x^{2}-60=0 \]

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