Solve (-28)2-7x/1-5x=3+7x/4+5x

$$( - 28 ) \frac { 2 - 7 x } { 1 - 5 x } = \frac { 3 + 7 x } { 4 + 5 x }$$

Solve for x

$$x=\frac{\sqrt{295941}-256}{1015}\approx 0.28374837$$

$$x=\frac{-\sqrt{295941}-256}{1015}\approx -0.788181868$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{4}{5},\frac{1}{5} since division by zero is not defined. Multiply both sides of the equation by \left(5x-1\right)\left(5x+4\right), the least common multiple of 1-5x,4+5x.

$$-28\left(-4-5x\right)\left(2-7x\right)=\left(5x-1\right)\left(3+7x\right)$$

Use the distributive property to multiply -28 by -4-5x.

$$\left(112+140x\right)\left(2-7x\right)=\left(5x-1\right)\left(3+7x\right)$$

Use the distributive property to multiply 112+140x by 2-7x and combine like terms.

$$224-504x-980x^{2}=\left(5x-1\right)\left(3+7x\right)$$

Use the distributive property to multiply 5x-1 by 3+7x and combine like terms.

$$224-504x-980x^{2}=8x+35x^{2}-3$$

Subtract 8x from both sides.

$$224-504x-980x^{2}-8x=35x^{2}-3$$

Combine -504x and -8x to get -512x.

$$224-512x-980x^{2}=35x^{2}-3$$

Subtract 35x^{2} from both sides.

$$224-512x-980x^{2}-35x^{2}=-3$$

Combine -980x^{2} and -35x^{2} to get -1015x^{2}.

$$224-512x-1015x^{2}=-3$$

Add 3 to both sides.

$$224-512x-1015x^{2}+3=0$$

Add 224 and 3 to get 227.

$$227-512x-1015x^{2}=0$$

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.

$$-1015x^{2}-512x+227=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1015 for a, -512 for b, and 227 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-512\right)±\sqrt{\left(-512\right)^{2}-4\left(-1015\right)\times 227}}{2\left(-1015\right)}$$

Square -512.

$$x=\frac{-\left(-512\right)±\sqrt{262144-4\left(-1015\right)\times 227}}{2\left(-1015\right)}$$

Multiply -4 times -1015.

$$x=\frac{-\left(-512\right)±\sqrt{262144+4060\times 227}}{2\left(-1015\right)}$$

Multiply 4060 times 227.

$$x=\frac{-\left(-512\right)±\sqrt{262144+921620}}{2\left(-1015\right)}$$

Add 262144 to 921620.

$$x=\frac{-\left(-512\right)±\sqrt{1183764}}{2\left(-1015\right)}$$

Take the square root of 1183764.

$$x=\frac{-\left(-512\right)±2\sqrt{295941}}{2\left(-1015\right)}$$

The opposite of -512 is 512.

$$x=\frac{512±2\sqrt{295941}}{2\left(-1015\right)}$$

Multiply 2 times -1015.

$$x=\frac{512±2\sqrt{295941}}{-2030}$$

Now solve the equation x=\frac{512±2\sqrt{295941}}{-2030} when ± is plus. Add 512 to 2\sqrt{295941}.

$$x=\frac{2\sqrt{295941}+512}{-2030}$$

Divide 512+2\sqrt{295941} by -2030.

$$x=\frac{-\sqrt{295941}-256}{1015}$$

Now solve the equation x=\frac{512±2\sqrt{295941}}{-2030} when ± is minus. Subtract 2\sqrt{295941} from 512.

$$x=\frac{512-2\sqrt{295941}}{-2030}$$

Divide 512-2\sqrt{295941} by -2030.

$$x=\frac{\sqrt{295941}-256}{1015}$$

The equation is now solved.

$$x=\frac{-\sqrt{295941}-256}{1015} x=\frac{\sqrt{295941}-256}{1015}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{4}{5},\frac{1}{5} since division by zero is not defined. Multiply both sides of the equation by \left(5x-1\right)\left(5x+4\right), the least common multiple of 1-5x,4+5x.

$$-28\left(-4-5x\right)\left(2-7x\right)=\left(5x-1\right)\left(3+7x\right)$$

Use the distributive property to multiply -28 by -4-5x.

$$\left(112+140x\right)\left(2-7x\right)=\left(5x-1\right)\left(3+7x\right)$$

Use the distributive property to multiply 112+140x by 2-7x and combine like terms.

$$224-504x-980x^{2}=\left(5x-1\right)\left(3+7x\right)$$

Use the distributive property to multiply 5x-1 by 3+7x and combine like terms.

$$224-504x-980x^{2}=8x+35x^{2}-3$$

Subtract 8x from both sides.

$$224-504x-980x^{2}-8x=35x^{2}-3$$

Combine -504x and -8x to get -512x.

$$224-512x-980x^{2}=35x^{2}-3$$

Subtract 35x^{2} from both sides.

$$224-512x-980x^{2}-35x^{2}=-3$$

Combine -980x^{2} and -35x^{2} to get -1015x^{2}.

$$224-512x-1015x^{2}=-3$$

Subtract 224 from both sides.

$$-512x-1015x^{2}=-3-224$$

Subtract 224 from -3 to get -227.

$$-512x-1015x^{2}=-227$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-1015x^{2}-512x=-227$$

Divide both sides by -1015.

$$\frac{-1015x^{2}-512x}{-1015}=-\frac{227}{-1015}$$

Dividing by -1015 undoes the multiplication by -1015.

$$x^{2}+\left(-\frac{512}{-1015}\right)x=-\frac{227}{-1015}$$

Divide -512 by -1015.

$$x^{2}+\frac{512}{1015}x=-\frac{227}{-1015}$$

Divide -227 by -1015.

$$x^{2}+\frac{512}{1015}x=\frac{227}{1015}$$

Divide \frac{512}{1015}, the coefficient of the x term, by 2 to get \frac{256}{1015}. Then add the square of \frac{256}{1015} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{512}{1015}x+\left(\frac{256}{1015}\right)^{2}=\frac{227}{1015}+\left(\frac{256}{1015}\right)^{2}$$

Square \frac{256}{1015} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{512}{1015}x+\frac{65536}{1030225}=\frac{227}{1015}+\frac{65536}{1030225}$$

Add \frac{227}{1015} to \frac{65536}{1030225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{512}{1015}x+\frac{65536}{1030225}=\frac{295941}{1030225}$$

Factor x^{2}+\frac{512}{1015}x+\frac{65536}{1030225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{256}{1015}\right)^{2}=\frac{295941}{1030225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{256}{1015}\right)^{2}}=\sqrt{\frac{295941}{1030225}}$$

Simplify.

$$x+\frac{256}{1015}=\frac{\sqrt{295941}}{1015} x+\frac{256}{1015}=-\frac{\sqrt{295941}}{1015}$$

Subtract \frac{256}{1015} from both sides of the equation.

$$x=\frac{\sqrt{295941}-256}{1015} x=\frac{-\sqrt{295941}-256}{1015}$$