Solve 5(19-x)/(x+4)(4-x)=2/3

$$\frac { 5 ( 19 - x ) } { ( x + 4 ) ( 4 - x ) } = \frac { 2 } { 3 }$$

Solve for x (complex solution)

$$x=\frac{15+\sqrt{1799}i}{4}\approx 3.75+10.60365503i$$

$$x=\frac{-\sqrt{1799}i+15}{4}\approx 3.75-10.60365503i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,4 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-4\right)\left(x+4\right), the least common multiple of \left(x+4\right)\left(4-x\right),3.

$$-3\times 5\left(19-x\right)=2\left(x-4\right)\left(x+4\right)$$

Multiply -3 and 5 to get -15.

$$-15\left(19-x\right)=2\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply -15 by 19-x.

$$-285+15x=2\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply 2 by x-4.

$$-285+15x=\left(2x-8\right)\left(x+4\right)$$

Use the distributive property to multiply 2x-8 by x+4 and combine like terms.

$$-285+15x=2x^{2}-32$$

Subtract 2x^{2} from both sides.

$$-285+15x-2x^{2}=-32$$

Add 32 to both sides.

$$-285+15x-2x^{2}+32=0$$

Add -285 and 32 to get -253.

$$-253+15x-2x^{2}=0$$

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.

$$-2x^{2}+15x-253=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 15 for b, and -253 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-2\right)\left(-253\right)}}{2\left(-2\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-2\right)\left(-253\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-15±\sqrt{225+8\left(-253\right)}}{2\left(-2\right)}$$

Multiply 8 times -253.

$$x=\frac{-15±\sqrt{225-2024}}{2\left(-2\right)}$$

Add 225 to -2024.

$$x=\frac{-15±\sqrt{-1799}}{2\left(-2\right)}$$

Take the square root of -1799.

$$x=\frac{-15±\sqrt{1799}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-15±\sqrt{1799}i}{-4}$$

Now solve the equation x=\frac{-15±\sqrt{1799}i}{-4} when ± is plus. Add -15 to i\sqrt{1799}.

$$x=\frac{-15+\sqrt{1799}i}{-4}$$

Divide -15+i\sqrt{1799} by -4.

$$x=\frac{-\sqrt{1799}i+15}{4}$$

Now solve the equation x=\frac{-15±\sqrt{1799}i}{-4} when ± is minus. Subtract i\sqrt{1799} from -15.

$$x=\frac{-\sqrt{1799}i-15}{-4}$$

Divide -15-i\sqrt{1799} by -4.

$$x=\frac{15+\sqrt{1799}i}{4}$$

The equation is now solved.

$$x=\frac{-\sqrt{1799}i+15}{4} x=\frac{15+\sqrt{1799}i}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,4 since division by zero is not defined. Multiply both sides of the equation by 3\left(x-4\right)\left(x+4\right), the least common multiple of \left(x+4\right)\left(4-x\right),3.

$$-3\times 5\left(19-x\right)=2\left(x-4\right)\left(x+4\right)$$

Multiply -3 and 5 to get -15.

$$-15\left(19-x\right)=2\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply -15 by 19-x.

$$-285+15x=2\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply 2 by x-4.

$$-285+15x=\left(2x-8\right)\left(x+4\right)$$

Use the distributive property to multiply 2x-8 by x+4 and combine like terms.

$$-285+15x=2x^{2}-32$$

Subtract 2x^{2} from both sides.

$$-285+15x-2x^{2}=-32$$

Add 285 to both sides.

$$15x-2x^{2}=-32+285$$

Add -32 and 285 to get 253.

$$15x-2x^{2}=253$$

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.

$$-2x^{2}+15x=253$$

Divide both sides by -2.

$$\frac{-2x^{2}+15x}{-2}=\frac{253}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{15}{-2}x=\frac{253}{-2}$$

Divide 15 by -2.

$$x^{2}-\frac{15}{2}x=\frac{253}{-2}$$

Divide 253 by -2.

$$x^{2}-\frac{15}{2}x=-\frac{253}{2}$$

Divide -\frac{15}{2}, the coefficient of the x term, by 2 to get -\frac{15}{4}. Then add the square of -\frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{2}x+\left(-\frac{15}{4}\right)^{2}=-\frac{253}{2}+\left(-\frac{15}{4}\right)^{2}$$

Square -\frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=-\frac{253}{2}+\frac{225}{16}$$

Add -\frac{253}{2} to \frac{225}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=-\frac{1799}{16}$$

Factor x^{2}-\frac{15}{2}x+\frac{225}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{4}\right)^{2}=-\frac{1799}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{4}\right)^{2}}=\sqrt{-\frac{1799}{16}}$$

Simplify.

$$x-\frac{15}{4}=\frac{\sqrt{1799}i}{4} x-\frac{15}{4}=-\frac{\sqrt{1799}i}{4}$$

Add \frac{15}{4} to both sides of the equation.

$$x=\frac{15+\sqrt{1799}i}{4} x=\frac{-\sqrt{1799}i+15}{4}$$