Solve x^2+3x-1/x+4=16-1-3x/x+4

$$x ^ { 2 } + \frac { 3 x - 1 } { x + 4 } = 16 - \frac { 1 - 3 x } { x + 4 }$$

Solve for x

$$x=4$$

Solution Steps

Variable x cannot be equal to -4 since division by zero is not defined. Multiply both sides of the equation by x+4.

$$\left(x+4\right)x^{2}+3x-1=\left(x+4\right)\times 16-\left(1-3x\right)$$

Use the distributive property to multiply x+4 by x^{2}.

$$x^{3}+4x^{2}+3x-1=\left(x+4\right)\times 16-\left(1-3x\right)$$

Use the distributive property to multiply x+4 by 16.

$$x^{3}+4x^{2}+3x-1=16x+64-\left(1-3x\right)$$

To find the opposite of 1-3x, find the opposite of each term.

$$x^{3}+4x^{2}+3x-1=16x+64-1+3x$$

Subtract 1 from 64 to get 63.

$$x^{3}+4x^{2}+3x-1=16x+63+3x$$

Combine 16x and 3x to get 19x.

$$x^{3}+4x^{2}+3x-1=19x+63$$

Subtract 19x from both sides.

$$x^{3}+4x^{2}+3x-1-19x=63$$

Combine 3x and -19x to get -16x.

$$x^{3}+4x^{2}-16x-1=63$$

Subtract 63 from both sides.

$$x^{3}+4x^{2}-16x-1-63=0$$

Subtract 63 from -1 to get -64.

$$x^{3}+4x^{2}-16x-64=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -64 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±64,±32,±16,±8,±4,±2,±1$$

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.

$$x=4$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}+4x^{2}-16x-64 by x-4 to get x^{2}+8x+16. Solve the equation where the result equals to 0.

$$x^{2}+8x+16=0$$

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, 8 for b, and 16 for c in the quadratic formula.

$$x=\frac{-8±\sqrt{8^{2}-4\times 1\times 16}}{2}$$

Do the calculations.

$$x=\frac{-8±0}{2}$$

Solutions are the same.

$$x=-4$$

Remove the values that the variable cannot be equal to.

$$x=4$$

List all found solutions.

$$x=4 x=-4$$

Variable x cannot be equal to -4.

$$x=4$$