Questions: Solve: p^4 + 3p^2 = 28

Transcript text: Solve: $p^{4}+3 p^{2}=28$

Solution

Step 1: Standard Form of the Equation

We start with the equation:

\[ p^{4} + 3p^{2} = 28 \]

By substituting $ u = p^{2} $, we rewrite the equation in standard form:

\[ u^{2} + 3u - 28 = 0 \]

Step 2: Factorization

Next, we factor the polynomial $ u^{2} + 3u - 28 $:

\[ (u - 4)(u + 7) = 0 \]

Step 3: Solutions for $ u $

Setting each factor to zero gives us the solutions for $ u $:

\[ u - 4 = 0 \quad \Rightarrow \quad u = 4 \] \[ u + 7 = 0 \quad \Rightarrow \quad u = -7 \]

Step 4: Solutions for $ p $

Now, we solve for $ p $ using the values of $ u $:

The solutions for $ p $ are:

\[ p = -2, \quad p = 2, \quad p = -\sqrt{7}i, \quad p = \sqrt{7}i \]

Thus, the final answer is:

\[ \boxed{p = -2, \quad p = 2, \quad p = -\sqrt{7}i, \quad p = \sqrt{7}i} \]

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