Solve 84/x+2-70/x=2/3

$$\frac { 84 } { x + 2 } - \frac { 70 } { x } = \frac { 2 } { 3 }$$

Solve for x (complex solution)

$$x=\frac{19+\sqrt{479}i}{2}\approx 9.5+10.943034314i$$

$$x=\frac{-\sqrt{479}i+19}{2}\approx 9.5-10.943034314i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+2\right), the least common multiple of x+2,x,3.

$$3x\times 84-\left(3x+6\right)\times 70=2x\left(x+2\right)$$

Multiply 3 and 84 to get 252.

$$252x-\left(3x+6\right)\times 70=2x\left(x+2\right)$$

Use the distributive property to multiply 3x+6 by 70.

$$252x-\left(210x+420\right)=2x\left(x+2\right)$$

To find the opposite of 210x+420, find the opposite of each term.

$$252x-210x-420=2x\left(x+2\right)$$

Combine 252x and -210x to get 42x.

$$42x-420=2x\left(x+2\right)$$

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

$$42x-420=2x^{2}+4x$$

Subtract 2x^{2} from both sides.

$$42x-420-2x^{2}=4x$$

Subtract 4x from both sides.

$$42x-420-2x^{2}-4x=0$$

Combine 42x and -4x to get 38x.

$$38x-420-2x^{2}=0$$

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.

$$-2x^{2}+38x-420=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 38 for b, and -420 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-38±\sqrt{38^{2}-4\left(-2\right)\left(-420\right)}}{2\left(-2\right)}$$

Square 38.

$$x=\frac{-38±\sqrt{1444-4\left(-2\right)\left(-420\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-38±\sqrt{1444+8\left(-420\right)}}{2\left(-2\right)}$$

Multiply 8 times -420.

$$x=\frac{-38±\sqrt{1444-3360}}{2\left(-2\right)}$$

Add 1444 to -3360.

$$x=\frac{-38±\sqrt{-1916}}{2\left(-2\right)}$$

Take the square root of -1916.

$$x=\frac{-38±2\sqrt{479}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-38±2\sqrt{479}i}{-4}$$

Now solve the equation x=\frac{-38±2\sqrt{479}i}{-4} when ± is plus. Add -38 to 2i\sqrt{479}.

$$x=\frac{-38+2\sqrt{479}i}{-4}$$

Divide -38+2i\sqrt{479} by -4.

$$x=\frac{-\sqrt{479}i+19}{2}$$

Now solve the equation x=\frac{-38±2\sqrt{479}i}{-4} when ± is minus. Subtract 2i\sqrt{479} from -38.

$$x=\frac{-2\sqrt{479}i-38}{-4}$$

Divide -38-2i\sqrt{479} by -4.

$$x=\frac{19+\sqrt{479}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{479}i+19}{2} x=\frac{19+\sqrt{479}i}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,0 since division by zero is not defined. Multiply both sides of the equation by 3x\left(x+2\right), the least common multiple of x+2,x,3.

$$3x\times 84-\left(3x+6\right)\times 70=2x\left(x+2\right)$$

Multiply 3 and 84 to get 252.

$$252x-\left(3x+6\right)\times 70=2x\left(x+2\right)$$

Use the distributive property to multiply 3x+6 by 70.

$$252x-\left(210x+420\right)=2x\left(x+2\right)$$

To find the opposite of 210x+420, find the opposite of each term.

$$252x-210x-420=2x\left(x+2\right)$$

Combine 252x and -210x to get 42x.

$$42x-420=2x\left(x+2\right)$$

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

$$42x-420=2x^{2}+4x$$

Subtract 2x^{2} from both sides.

$$42x-420-2x^{2}=4x$$

Subtract 4x from both sides.

$$42x-420-2x^{2}-4x=0$$

Combine 42x and -4x to get 38x.

$$38x-420-2x^{2}=0$$

Add 420 to both sides. Anything plus zero gives itself.

$$38x-2x^{2}=420$$

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.

$$-2x^{2}+38x=420$$

Divide both sides by -2.

$$\frac{-2x^{2}+38x}{-2}=\frac{420}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{38}{-2}x=\frac{420}{-2}$$

Divide 38 by -2.

$$x^{2}-19x=\frac{420}{-2}$$

Divide 420 by -2.

$$x^{2}-19x=-210$$

Divide -19, the coefficient of the x term, by 2 to get -\frac{19}{2}. Then add the square of -\frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=-210+\left(-\frac{19}{2}\right)^{2}$$

Square -\frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-19x+\frac{361}{4}=-210+\frac{361}{4}$$

Add -210 to \frac{361}{4}.

$$x^{2}-19x+\frac{361}{4}=-\frac{479}{4}$$

Factor x^{2}-19x+\frac{361}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{2}\right)^{2}=-\frac{479}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{-\frac{479}{4}}$$

Simplify.

$$x-\frac{19}{2}=\frac{\sqrt{479}i}{2} x-\frac{19}{2}=-\frac{\sqrt{479}i}{2}$$

Add \frac{19}{2} to both sides of the equation.

$$x=\frac{19+\sqrt{479}i}{2} x=\frac{-\sqrt{479}i+19}{2}$$