Factor and solve for $$ x $$. $$ x^{4}-30 x^{2}+25=0 $$

Question

UpStudy AI · Accepted Answer

We start by letting
$$
y = x^2.
$$
Then the equation
$$
x^{4} - 30x^2 + 25 = 0
$$
becomes
$$
y^2 - 30y + 25 = 0.
$$

### Step 1. Solve the Quadratic in $$ y $$
We use the quadratic formula:
$$
y = \frac{30 \pm \sqrt{30^2 - 4 \times 1 \times 25}}{2 \times 1}.
$$
Simplify inside the square root:
$$
30^2 - 4 \times 25 = 900 - 100 = 800.
$$
Thus,
$$
y = \frac{30 \pm \sqrt{800}}{2}.
$$
Notice that
$$
\sqrt{800} = \sqrt{16 \times 50} = 4\sqrt{50} = 4 \times 5\sqrt{2} = 20\sqrt{2}.
$$
So,
$$
y = \frac{30 \pm 20\sqrt{2}}{2} = 15 \pm 10\sqrt{2}.
$$

### Step 2. Substitute Back for $$ x $$
Recall
$$
y = x^2.
$$
Thus, we have two equations:
$$
x^2 = 15 + 10\sqrt{2} \quad \text{and} \quad x^2 = 15 - 10\sqrt{2}.
$$
Taking the square root of both sides gives:

For $$ x^2 = 15 + 10\sqrt{2} $$,
$$
x = \sqrt{15 + 10\sqrt{2}} \quad \text{or} \quad x = -\sqrt{15 + 10\sqrt{2}},
$$
and for $$ x^2 = 15 - 10\sqrt{2} $$,
$$
x = \sqrt{15 - 10\sqrt{2}} \quad \text{or} \quad x = -\sqrt{15 - 10\sqrt{2}}.
$$

### Final Answer
The solutions to the equation $$ x^{4} - 30x^{2} + 25 = 0 $$ are:
$$
x = \pm\sqrt{15 + 10\sqrt{2}} \quad \text{and} \quad x = \pm\sqrt{15 - 10\sqrt{2}}.
$$
The solutions are $$ x = \sqrt{15 + 10\sqrt{2}} $$, $$ x = -\sqrt{15 + 10\sqrt{2}} $$, $$ x = \sqrt{15 - 10\sqrt{2}} $$, and $$ x = -\sqrt{15 - 10\sqrt{2}} $$.

Factor and solve for .

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