Question

Solve each equation, if possible. $$ p\left(p^{2}+3\right)=12+p^{3} $$

Short answer

The solution is \( p = 4 \).

Step-by-step solution

Rearrange the equation

Start by rewriting the given equation in a standard form where all terms are on one side. The given equation is: \[ p(p^2 + 3) = 12 + p^3 \] Expand the left side: \[ p^3 + 3p = 12 + p^3 \] Subtract \( p^3 \) from both sides to get: \[ 3p = 12 \]

Isolate the variable

Now, solve for \( p \) by dividing both sides of the equation by 3: \[ p = \frac{12}{3} \] Simplify the fraction: \[ p = 4 \]

Verify the solution

Substitute \( p = 4 \) back into the original equation to verify the solution: \[ 4(4^2 + 3) = 12 + 4^3 \] Simplify inside the parentheses: \[ 4(16 + 3) = 12 + 64 \] Further simplify: \[ 4 \times 19 = 76 \] and \[ 12 + 64 = 76 \] Both sides are equal, confirming that \( p = 4 \) is a valid solution.

Key concepts

Rearranging Equations

Rearranging equations is often the first step in solving them. It's crucial because it helps to get all terms involving the variable on one side of the equation. Consider the given equation: \[ p(p^2 + 3) = 12 + p^3 \] We start by expanding the left side: \[ p^3 + 3p \] Now, rewrite the equation as: \[ p^3 + 3p = 12 + p^3 \] To rearrange it into a standard form, subtract \( p^3 \) from both sides to get: \[ 3p = 12 \] This form, \( 3p = 12 \), makes the next step, isolating the variable, much easier.

Isolating the Variable

Once the equation is rearranged, the next step is to isolate the variable. Isolating the variable means getting the variable by itself on one side of the equation. Starting from: \[ 3p = 12 \] We isolate \( p \) by dividing both sides by 3: \[ p = \frac{12}{3} \] This simplifies to: \[ p = 4 \] Now \( p \) is isolated, meaning we have found the value of the variable. This value can then be used in the final step to verify the solution.

Verifying Solutions

Verifying the solution is the process of checking that the calculated value satisfies the original equation. It's an essential step to ensure there were no mistakes in solving. Substitute \( p = 4 \) back into the original equation: \[ 4(4^2 + 3) = 12 + 4^3 \] Simplify inside the parentheses: \[ 4(16 + 3) = 12 + 64 \] Further simplify both sides: \[ 4 \times 19 = 76 \] and \[ 12 + 64 = 76 \] Both sides are equal, confirming that \( p = 4 \) is a valid solution. This process validates our previous work and assures us that the solution is correct.