Solve 7/x+5.5/x-3=1

$$\frac{ 7 }{ x } + \frac{ 5.5 }{ x-3 } = 1$$

Solve for x

$$x=14$$

$$x=1.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x,x-3.

$$\left(x-3\right)\times 7+x\times 5.5=x\left(x-3\right)$$

Use the distributive property to multiply x-3 by 7.

$$7x-21+x\times 5.5=x\left(x-3\right)$$

Combine 7x and x\times 5.5 to get 12.5x.

$$12.5x-21=x\left(x-3\right)$$

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

$$12.5x-21=x^{2}-3x$$

Subtract x^{2} from both sides.

$$12.5x-21-x^{2}=-3x$$

Add 3x to both sides.

$$12.5x-21-x^{2}+3x=0$$

Combine 12.5x and 3x to get 15.5x.

$$15.5x-21-x^{2}=0$$

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.

$$-x^{2}+15.5x-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 15.5 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15.5±\sqrt{15.5^{2}-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}$$

Square 15.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-15.5±\sqrt{240.25-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-15.5±\sqrt{240.25+4\left(-21\right)}}{2\left(-1\right)}$$

Multiply 4 times -21.

$$x=\frac{-15.5±\sqrt{240.25-84}}{2\left(-1\right)}$$

Add 240.25 to -84.

$$x=\frac{-15.5±\sqrt{156.25}}{2\left(-1\right)}$$

Take the square root of 156.25.

$$x=\frac{-15.5±\frac{25}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-15.5±\frac{25}{2}}{-2}$$

Now solve the equation x=\frac{-15.5±\frac{25}{2}}{-2} when ± is plus. Add -15.5 to \frac{25}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{3}{-2}$$

Divide -3 by -2.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{-15.5±\frac{25}{2}}{-2} when ± is minus. Subtract \frac{25}{2} from -15.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{28}{-2}$$

Divide -28 by -2.

$$x=14$$

The equation is now solved.

$$x=\frac{3}{2} x=14$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x,x-3.

$$\left(x-3\right)\times 7+x\times 5.5=x\left(x-3\right)$$

Use the distributive property to multiply x-3 by 7.

$$7x-21+x\times 5.5=x\left(x-3\right)$$

Combine 7x and x\times 5.5 to get 12.5x.

$$12.5x-21=x\left(x-3\right)$$

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

$$12.5x-21=x^{2}-3x$$

Subtract x^{2} from both sides.

$$12.5x-21-x^{2}=-3x$$

Add 3x to both sides.

$$12.5x-21-x^{2}+3x=0$$

Combine 12.5x and 3x to get 15.5x.

$$15.5x-21-x^{2}=0$$

Add 21 to both sides. Anything plus zero gives itself.

$$15.5x-x^{2}=21$$

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.

$$-x^{2}+15.5x=21$$

Divide both sides by -1.

$$\frac{-x^{2}+15.5x}{-1}=\frac{21}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{15.5}{-1}x=\frac{21}{-1}$$

Divide 15.5 by -1.

$$x^{2}-15.5x=\frac{21}{-1}$$

Divide 21 by -1.

$$x^{2}-15.5x=-21$$

Divide -15.5, the coefficient of the x term, by 2 to get -7.75. Then add the square of -7.75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15.5x+\left(-7.75\right)^{2}=-21+\left(-7.75\right)^{2}$$

Square -7.75 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15.5x+60.0625=-21+60.0625$$

Add -21 to 60.0625.

$$x^{2}-15.5x+60.0625=39.0625$$

Factor x^{2}-15.5x+60.0625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7.75\right)^{2}=39.0625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7.75\right)^{2}}=\sqrt{39.0625}$$

Simplify.

$$x-7.75=\frac{25}{4} x-7.75=-\frac{25}{4}$$

Add 7.75 to both sides of the equation.

$$x=14 x=\frac{3}{2}$$