Solve left(-2x+1right)x+45/x^2-10x+25+3/x-5=0

$$\frac{ \left( -2x+1 \right) x+45 }{ { x }^{ 2 } -10x+25 } + \frac{ 3 }{ x-5 } = 0$$

Solve for x

$$x=-3$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)^{2}, the least common multiple of x^{2}-10x+25,x-5.

$$\left(-2x+1\right)x+45+\left(x-5\right)\times 3=0$$

Use the distributive property to multiply -2x+1 by x.

$$-2x^{2}+x+45+\left(x-5\right)\times 3=0$$

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

$$-2x^{2}+x+45+3x-15=0$$

Combine x and 3x to get 4x.

$$-2x^{2}+4x+45-15=0$$

Subtract 15 from 45 to get 30.

$$-2x^{2}+4x+30=0$$

Divide both sides by 2.

$$-x^{2}+2x+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=2 ab=-15=-15$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 2.

$$a=5 b=-3$$

Rewrite -x^{2}+2x+15 as \left(-x^{2}+5x\right)+\left(-3x+15\right).

$$\left(-x^{2}+5x\right)+\left(-3x+15\right)$$

Factor out -x in the first and -3 in the second group.

$$-x\left(x-5\right)-3\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x-3\right)$$

To find equation solutions, solve x-5=0 and -x-3=0.

$$x=5 x=-3$$

Variable x cannot be equal to 5.

$$x=-3$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)^{2}, the least common multiple of x^{2}-10x+25,x-5.

$$\left(-2x+1\right)x+45+\left(x-5\right)\times 3=0$$

Use the distributive property to multiply -2x+1 by x.

$$-2x^{2}+x+45+\left(x-5\right)\times 3=0$$

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

$$-2x^{2}+x+45+3x-15=0$$

Combine x and 3x to get 4x.

$$-2x^{2}+4x+45-15=0$$

Subtract 15 from 45 to get 30.

$$-2x^{2}+4x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 4 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-2\right)\times 30}}{2\left(-2\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)\times 30}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-4±\sqrt{16+8\times 30}}{2\left(-2\right)}$$

Multiply 8 times 30.

$$x=\frac{-4±\sqrt{16+240}}{2\left(-2\right)}$$

Add 16 to 240.

$$x=\frac{-4±\sqrt{256}}{2\left(-2\right)}$$

Take the square root of 256.

$$x=\frac{-4±16}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-4±16}{-4}$$

Now solve the equation x=\frac{-4±16}{-4} when ± is plus. Add -4 to 16.

$$x=\frac{12}{-4}$$

Divide 12 by -4.

$$x=-3$$

Now solve the equation x=\frac{-4±16}{-4} when ± is minus. Subtract 16 from -4.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

The equation is now solved.

$$x=-3 x=5$$

Variable x cannot be equal to 5.

$$x=-3$$

Steps for Completing the Square

Variable x cannot be equal to 5 since division by zero is not defined. Multiply both sides of the equation by \left(x-5\right)^{2}, the least common multiple of x^{2}-10x+25,x-5.

$$\left(-2x+1\right)x+45+\left(x-5\right)\times 3=0$$

Use the distributive property to multiply -2x+1 by x.

$$-2x^{2}+x+45+\left(x-5\right)\times 3=0$$

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

$$-2x^{2}+x+45+3x-15=0$$

Combine x and 3x to get 4x.

$$-2x^{2}+4x+45-15=0$$

Subtract 15 from 45 to get 30.

$$-2x^{2}+4x+30=0$$

Subtract 30 from both sides. Anything subtracted from zero gives its negation.

$$-2x^{2}+4x=-30$$

Divide both sides by -2.

$$\frac{-2x^{2}+4x}{-2}=-\frac{30}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{4}{-2}x=-\frac{30}{-2}$$

Divide 4 by -2.

$$x^{2}-2x=-\frac{30}{-2}$$

Divide -30 by -2.

$$x^{2}-2x=15$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=15+1$$

Add 15 to 1.

$$x^{2}-2x+1=16$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-1=4 x-1=-4$$

Add 1 to both sides of the equation.

$$x=5 x=-3$$

Variable x cannot be equal to 5.

$$x=-3$$