Solve y^2=48*36/y

$${ y }^{ 2 } = \frac{ 48 \times 36 }{ y }$$

Solve for y

$$y=12$$

Solution Steps

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$yy^{2}=48\times 36$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$y^{3}=48\times 36$$

Multiply 48 and 36 to get 1728.

$$y^{3}=1728$$

Subtract 1728 from both sides.

$$y^{3}-1728=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -1728 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±1728,±864,±576,±432,±288,±216,±192,±144,±108,±96,±72,±64,±54,±48,±36,±32,±27,±24,±18,±16,±12,±9,±8,±6,±4,±3,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$y=12$$

By Factor theorem, y-k is a factor of the polynomial for each root k. Divide y^{3}-1728 by y-12 to get y^{2}+12y+144. Solve the equation where the result equals to 0.

$$y^{2}+12y+144=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 1 for a, 12 for b, and 144 for c in the quadratic formula.

$$y=\frac{-12±\sqrt{12^{2}-4\times 1\times 144}}{2}$$

Do the calculations.

$$y=\frac{-12±\sqrt{-432}}{2}$$

Since the square root of a negative number is not defined in the real field, there are no solutions.

$$y\in \emptyset$$

List all found solutions.

$$y=12$$