Solve x+19/25+6left(2+x/5right)left(1/5x-2right)=4+2left(x/5-6right)-frac{6}{25}+3(frac{x}{5}-4)

$$x+ \frac{ 19 }{ 25 } +6 \left( 2+ \frac{ x }{ 5 } \right) \left( \frac{ 1 }{ 5 } x-2 \right) = 4+2 \left( \frac{ x }{ 5 } -6 \right) - \frac{ 6 }{ 25 } +3( \frac{ x }{ 5 } -4)$$

Solve for x

$$x = \frac{5 \sqrt{2}}{2} \approx 3.535533906$$

$$x = -\frac{5 \sqrt{2}}{2} \approx -3.535533906$$

Steps by Finding Square Root

Multiply both sides of the equation by 25, the least common multiple of 25,5.

$$25x+19+150\left(2+\frac{x}{5}\right)\left(\frac{1}{5}x-2\right)=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Use the distributive property to multiply 150 by 2+\frac{x}{5}.

$$25x+19+\left(300+150\times \frac{x}{5}\right)\left(\frac{1}{5}x-2\right)=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Cancel out 5, the greatest common factor in 150 and 5.

$$25x+19+\left(300+30x\right)\left(\frac{1}{5}x-2\right)=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Apply the distributive property by multiplying each term of 300+30x by each term of \frac{1}{5}x-2.

$$25x+19+300\times \frac{1}{5}x-600+30x\times \frac{1}{5}x-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

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

$$25x+19+300\times \frac{1}{5}x-600+30x^{2}\times \frac{1}{5}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Multiply 300 and \frac{1}{5} to get \frac{300}{5}.

$$25x+19+\frac{300}{5}x-600+30x^{2}\times \frac{1}{5}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Divide 300 by 5 to get 60.

$$25x+19+60x-600+30x^{2}\times \frac{1}{5}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Multiply 30 and \frac{1}{5} to get \frac{30}{5}.

$$25x+19+60x-600+\frac{30}{5}x^{2}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Divide 30 by 5 to get 6.

$$25x+19+60x-600+6x^{2}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Combine 60x and -60x to get 0.

$$25x+19-600+6x^{2}=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Subtract 600 from 19 to get -581.

$$25x-581+6x^{2}=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Use the distributive property to multiply 50 by \frac{x}{5}-6.

$$25x-581+6x^{2}=100+50\times \frac{x}{5}-300-6+75\left(\frac{x}{5}-4\right)$$

Cancel out 5, the greatest common factor in 50 and 5.

$$25x-581+6x^{2}=100+10x-300-6+75\left(\frac{x}{5}-4\right)$$

Subtract 300 from 100 to get -200.

$$25x-581+6x^{2}=-200+10x-6+75\left(\frac{x}{5}-4\right)$$

Subtract 6 from -200 to get -206.

$$25x-581+6x^{2}=-206+10x+75\left(\frac{x}{5}-4\right)$$

Use the distributive property to multiply 75 by \frac{x}{5}-4.

$$25x-581+6x^{2}=-206+10x+75\times \frac{x}{5}-300$$

Cancel out 5, the greatest common factor in 75 and 5.

$$25x-581+6x^{2}=-206+10x+15x-300$$

Combine 10x and 15x to get 25x.

$$25x-581+6x^{2}=-206+25x-300$$

Subtract 300 from -206 to get -506.

$$25x-581+6x^{2}=-506+25x$$

Subtract 25x from both sides.

$$25x-581+6x^{2}-25x=-506$$

Combine 25x and -25x to get 0.

$$-581+6x^{2}=-506$$

Add 581 to both sides.

$$6x^{2}=-506+581$$

Add -506 and 581 to get 75.

$$6x^{2}=75$$

Divide both sides by 6.

$$x^{2}=\frac{75}{6}$$

Reduce the fraction \frac{75}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}=\frac{25}{2}$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Multiply both sides of the equation by 25, the least common multiple of 25,5.

$$25x+19+150\left(2+\frac{x}{5}\right)\left(\frac{1}{5}x-2\right)=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Use the distributive property to multiply 150 by 2+\frac{x}{5}.

$$25x+19+\left(300+150\times \frac{x}{5}\right)\left(\frac{1}{5}x-2\right)=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Cancel out 5, the greatest common factor in 150 and 5.

$$25x+19+\left(300+30x\right)\left(\frac{1}{5}x-2\right)=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Apply the distributive property by multiplying each term of 300+30x by each term of \frac{1}{5}x-2.

$$25x+19+300\times \frac{1}{5}x-600+30x\times \frac{1}{5}x-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

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

$$25x+19+300\times \frac{1}{5}x-600+30x^{2}\times \frac{1}{5}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Multiply 300 and \frac{1}{5} to get \frac{300}{5}.

$$25x+19+\frac{300}{5}x-600+30x^{2}\times \frac{1}{5}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Divide 300 by 5 to get 60.

$$25x+19+60x-600+30x^{2}\times \frac{1}{5}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Multiply 30 and \frac{1}{5} to get \frac{30}{5}.

$$25x+19+60x-600+\frac{30}{5}x^{2}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Divide 30 by 5 to get 6.

$$25x+19+60x-600+6x^{2}-60x=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Combine 60x and -60x to get 0.

$$25x+19-600+6x^{2}=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Subtract 600 from 19 to get -581.

$$25x-581+6x^{2}=100+50\left(\frac{x}{5}-6\right)-6+75\left(\frac{x}{5}-4\right)$$

Use the distributive property to multiply 50 by \frac{x}{5}-6.

$$25x-581+6x^{2}=100+50\times \frac{x}{5}-300-6+75\left(\frac{x}{5}-4\right)$$

Cancel out 5, the greatest common factor in 50 and 5.

$$25x-581+6x^{2}=100+10x-300-6+75\left(\frac{x}{5}-4\right)$$

Subtract 300 from 100 to get -200.

$$25x-581+6x^{2}=-200+10x-6+75\left(\frac{x}{5}-4\right)$$

Subtract 6 from -200 to get -206.

$$25x-581+6x^{2}=-206+10x+75\left(\frac{x}{5}-4\right)$$

Use the distributive property to multiply 75 by \frac{x}{5}-4.

$$25x-581+6x^{2}=-206+10x+75\times \frac{x}{5}-300$$

Cancel out 5, the greatest common factor in 75 and 5.

$$25x-581+6x^{2}=-206+10x+15x-300$$

Combine 10x and 15x to get 25x.

$$25x-581+6x^{2}=-206+25x-300$$

Subtract 300 from -206 to get -506.

$$25x-581+6x^{2}=-506+25x$$

Subtract -506 from both sides.

$$25x-581+6x^{2}-\left(-506\right)=25x$$

The opposite of -506 is 506.

$$25x-581+6x^{2}+506=25x$$

Subtract 25x from both sides.

$$25x-581+6x^{2}+506-25x=0$$

Add -581 and 506 to get -75.

$$25x-75+6x^{2}-25x=0$$

Combine 25x and -25x to get 0.

$$-75+6x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$6x^{2}-75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 0 for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 6\left(-75\right)}}{2\times 6}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 6\left(-75\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{0±\sqrt{-24\left(-75\right)}}{2\times 6}$$

Multiply -24 times -75.

$$x=\frac{0±\sqrt{1800}}{2\times 6}$$

Take the square root of 1800.

$$x=\frac{0±30\sqrt{2}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{0±30\sqrt{2}}{12}$$

Now solve the equation x=\frac{0±30\sqrt{2}}{12} when ± is plus.

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

Now solve the equation x=\frac{0±30\sqrt{2}}{12} when ± is minus.

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

The equation is now solved.

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