Question

determine the solution. $\frac{3}{x^{2}}=\frac{x + 4}{5x^{2}}$

Explanation:

Step1: Cross - multiply

Since $\frac{3}{x^{2}}=\frac{x + 4}{5x^{2}}$, cross - multiplying gives $3\times5x^{2}=(x + 4)\times x^{2}$.

Step2: Expand both sides

The left - hand side is $15x^{2}$, and the right - hand side is $x^{3}+4x^{2}$. So, $15x^{2}=x^{3}+4x^{2}$.

Step3: Rearrange the equation

Subtract $15x^{2}$ from both sides to get $0=x^{3}+4x^{2}-15x^{2}$, which simplifies to $x^{3}-11x^{2}=0$.

Step4: Factor out the common factor

Factor out $x^{2}$ from the left - hand side: $x^{2}(x - 11)=0$.

Step5: Solve for x

Using the zero - product property, if $ab = 0$, then either $a = 0$ or $b = 0$. So, $x^{2}=0$ gives $x = 0$, and $x-11=0$ gives $x = 11$. But when $x = 0$, the original equation $\frac{3}{x^{2}}=\frac{x + 4}{5x^{2}}$ has undefined terms (division by zero). So we discard $x = 0$.

Answer:

$x = 11$