Solve x/5+5/x=41/4

$$\frac { x } { 5 } + \frac { 5 } { x } = 4 \frac { 1 } { 4 }$$

Solve for x

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

$$x=20$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 20x, the least common multiple of 5,x,4.

$$4xx+20\times 5=5x\left(4\times 4+1\right)$$

Multiply x and x to get x^{2}.

$$4x^{2}+20\times 5=5x\left(4\times 4+1\right)$$

Multiply 20 and 5 to get 100.

$$4x^{2}+100=5x\left(4\times 4+1\right)$$

Multiply 4 and 4 to get 16.

$$4x^{2}+100=5x\left(16+1\right)$$

Add 16 and 1 to get 17.

$$4x^{2}+100=5x\times 17$$

Multiply 5 and 17 to get 85.

$$4x^{2}+100=85x$$

Subtract 85x from both sides.

$$4x^{2}+100-85x=0$$

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

$$4x^{2}-85x+100=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx+100. To find a and b, set up a system to be solved.

$$a+b=-85 ab=4\times 100=400$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 400.

$$-1,-400 -2,-200 -4,-100 -5,-80 -8,-50 -10,-40 -16,-25 -20,-20$$

Calculate the sum for each pair.

$$-1-400=-401 -2-200=-202 -4-100=-104 -5-80=-85 -8-50=-58 -10-40=-50 -16-25=-41 -20-20=-40$$

The solution is the pair that gives sum -85.

$$a=-80 b=-5$$

Rewrite 4x^{2}-85x+100 as \left(4x^{2}-80x\right)+\left(-5x+100\right).

$$\left(4x^{2}-80x\right)+\left(-5x+100\right)$$

Factor out 4x in the first and -5 in the second group.

$$4x\left(x-20\right)-5\left(x-20\right)$$

Factor out common term x-20 by using distributive property.

$$\left(x-20\right)\left(4x-5\right)$$

To find equation solutions, solve x-20=0 and 4x-5=0.

$$x=20 x=\frac{5}{4}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 20x, the least common multiple of 5,x,4.

$$4xx+20\times 5=5x\left(4\times 4+1\right)$$

Multiply x and x to get x^{2}.

$$4x^{2}+20\times 5=5x\left(4\times 4+1\right)$$

Multiply 20 and 5 to get 100.

$$4x^{2}+100=5x\left(4\times 4+1\right)$$

Multiply 4 and 4 to get 16.

$$4x^{2}+100=5x\left(16+1\right)$$

Add 16 and 1 to get 17.

$$4x^{2}+100=5x\times 17$$

Multiply 5 and 17 to get 85.

$$4x^{2}+100=85x$$

Subtract 85x from both sides.

$$4x^{2}+100-85x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$4x^{2}-85x+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -85 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-85\right)±\sqrt{\left(-85\right)^{2}-4\times 4\times 100}}{2\times 4}$$

Square -85.

$$x=\frac{-\left(-85\right)±\sqrt{7225-4\times 4\times 100}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-85\right)±\sqrt{7225-16\times 100}}{2\times 4}$$

Multiply -16 times 100.

$$x=\frac{-\left(-85\right)±\sqrt{7225-1600}}{2\times 4}$$

Add 7225 to -1600.

$$x=\frac{-\left(-85\right)±\sqrt{5625}}{2\times 4}$$

Take the square root of 5625.

$$x=\frac{-\left(-85\right)±75}{2\times 4}$$

The opposite of -85 is 85.

$$x=\frac{85±75}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{85±75}{8}$$

Now solve the equation x=\frac{85±75}{8} when ± is plus. Add 85 to 75.

$$x=\frac{160}{8}$$

Divide 160 by 8.

$$x=20$$

Now solve the equation x=\frac{85±75}{8} when ± is minus. Subtract 75 from 85.

$$x=\frac{10}{8}$$

Reduce the fraction \frac{10}{8} to lowest terms by extracting and canceling out 2.

$$x=\frac{5}{4}$$

The equation is now solved.

$$x=20 x=\frac{5}{4}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 20x, the least common multiple of 5,x,4.

$$4xx+20\times 5=5x\left(4\times 4+1\right)$$

Multiply x and x to get x^{2}.

$$4x^{2}+20\times 5=5x\left(4\times 4+1\right)$$

Multiply 20 and 5 to get 100.

$$4x^{2}+100=5x\left(4\times 4+1\right)$$

Multiply 4 and 4 to get 16.

$$4x^{2}+100=5x\left(16+1\right)$$

Add 16 and 1 to get 17.

$$4x^{2}+100=5x\times 17$$

Multiply 5 and 17 to get 85.

$$4x^{2}+100=85x$$

Subtract 85x from both sides.

$$4x^{2}+100-85x=0$$

Subtract 100 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-85x=-100$$

Divide both sides by 4.

$$\frac{4x^{2}-85x}{4}=-\frac{100}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{85}{4}x=-\frac{100}{4}$$

Divide -100 by 4.

$$x^{2}-\frac{85}{4}x=-25$$

Divide -\frac{85}{4}, the coefficient of the x term, by 2 to get -\frac{85}{8}. Then add the square of -\frac{85}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{85}{4}x+\left(-\frac{85}{8}\right)^{2}=-25+\left(-\frac{85}{8}\right)^{2}$$

Square -\frac{85}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{85}{4}x+\frac{7225}{64}=-25+\frac{7225}{64}$$

Add -25 to \frac{7225}{64}.

$$x^{2}-\frac{85}{4}x+\frac{7225}{64}=\frac{5625}{64}$$

Factor x^{2}-\frac{85}{4}x+\frac{7225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{85}{8}\right)^{2}=\frac{5625}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{85}{8}\right)^{2}}=\sqrt{\frac{5625}{64}}$$

Simplify.

$$x-\frac{85}{8}=\frac{75}{8} x-\frac{85}{8}=-\frac{75}{8}$$

Add \frac{85}{8} to both sides of the equation.

$$x=20 x=\frac{5}{4}$$