Solve (x^2-5x+6/x-3)-1:(x^2-4/5)=0

$$( \frac { x ^ { 2 } - 5 x + 6 } { x - 3 } ) - 1 : ( \frac { x ^ { 2 } - 4 } { 5 } ) = 0$$

Solve for x

$$x=\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

$$x=\frac{-\sqrt{5}-1}{2}\approx -1.618033989$$

Solution Steps

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by x-3.

$$x^{2}-5x+6-5\left(x-3\right)\left(x^{2}-4\right)^{-1}=0$$

Use the distributive property to multiply -5 by x-3.

$$x^{2}-5x+6+\left(-5x+15\right)\left(x^{2}-4\right)^{-1}=0$$

Use the distributive property to multiply -5x+15 by \left(x^{2}-4\right)^{-1}.

$$x^{2}-5x+6-5x\left(x^{2}-4\right)^{-1}+15\left(x^{2}-4\right)^{-1}=0$$

Reorder the terms.

$$x^{2}-5x+6-5\times \frac{1}{x^{2}-4}x+15\times \frac{1}{x^{2}-4}=0$$

Variable x cannot be equal to any of the values -2,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x+2\right).

$$\left(x-2\right)\left(x+2\right)x^{2}-5x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 6-5x+15\times 1=0$$

Do the multiplications.

$$\left(x-2\right)\left(x+2\right)x^{2}-5x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 6-5x+15=0$$

Consider \left(x-2\right)\left(x+2\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 2.

$$\left(x^{2}-4\right)x^{2}-5x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 6-5x+15=0$$

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

$$x^{4}-4x^{2}-5x\left(x-2\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 6-5x+15=0$$

Use the distributive property to multiply -5x by x-2.

$$x^{4}-4x^{2}+\left(-5x^{2}+10x\right)\left(x+2\right)+\left(x-2\right)\left(x+2\right)\times 6-5x+15=0$$

Use the distributive property to multiply -5x^{2}+10x by x+2 and combine like terms.

$$x^{4}-4x^{2}-5x^{3}+20x+\left(x-2\right)\left(x+2\right)\times 6-5x+15=0$$

Use the distributive property to multiply x-2 by x+2 and combine like terms.

$$x^{4}-4x^{2}-5x^{3}+20x+\left(x^{2}-4\right)\times 6-5x+15=0$$

Use the distributive property to multiply x^{2}-4 by 6.

$$x^{4}-4x^{2}-5x^{3}+20x+6x^{2}-24-5x+15=0$$

Combine -4x^{2} and 6x^{2} to get 2x^{2}.

$$x^{4}+2x^{2}-5x^{3}+20x-24-5x+15=0$$

Combine 20x and -5x to get 15x.

$$x^{4}+2x^{2}-5x^{3}+15x-24+15=0$$

Add -24 and 15 to get -9.

$$x^{4}+2x^{2}-5x^{3}+15x-9=0$$

Rearrange the equation to put it in standard form. Place the terms in order from highest to lowest power.

$$x^{4}-5x^{3}+2x^{2}+15x-9=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term -9 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±9,±3,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{4}-5x^{3}+2x^{2}+15x-9 by x-3 to get x^{3}-2x^{2}-4x+3. Solve the equation where the result equals to 0.

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

By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 3 and q divides the leading coefficient 1. List all candidates \frac{p}{q}.

$$±3,±1$$

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.

$$x=3$$

By Factor theorem, x-k is a factor of the polynomial for each root k. Divide x^{3}-2x^{2}-4x+3 by x-3 to get x^{2}+x-1. Solve the equation where the result equals to 0.

$$x^{2}+x-1=0$$

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, 1 for b, and -1 for c in the quadratic formula.

$$x=\frac{-1±\sqrt{1^{2}-4\times 1\left(-1\right)}}{2}$$

Do the calculations.

$$x=\frac{-1±\sqrt{5}}{2}$$

Solve the equation x^{2}+x-1=0 when ± is plus and when ± is minus.

$$x=\frac{-\sqrt{5}-1}{2} x=\frac{\sqrt{5}-1}{2}$$

Remove the values that the variable cannot be equal to.

$$x\in \emptyset$$

List all found solutions.

$$x=3 x=\frac{-\sqrt{5}-1}{2} x=\frac{\sqrt{5}-1}{2}$$

Variable x cannot be equal to 3.

$$x=\frac{\sqrt{5}-1}{2} x=\frac{-\sqrt{5}-1}{2}$$