Solve 14*x/12+x*14/12+x=4

$$14 \times \frac{ x }{ 12+x } \times \frac{ 14 }{ 12+x } =4$$

Solve for x

$$x=9$$

$$x=16$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -12 since division by zero is not defined. Multiply both sides of the equation by x+12.

$$14x\times \frac{14}{12+x}=4\left(x+12\right)$$

Express 14\times \frac{14}{12+x} as a single fraction.

$$\frac{14\times 14}{12+x}x=4\left(x+12\right)$$

Use the distributive property to multiply 4 by x+12.

$$\frac{14\times 14}{12+x}x=4x+48$$

Multiply 14 and 14 to get 196.

$$\frac{196}{12+x}x=4x+48$$

Express \frac{196}{12+x}x as a single fraction.

$$\frac{196x}{12+x}=4x+48$$

Subtract 4x from both sides.

$$\frac{196x}{12+x}-4x=48$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -4x times \frac{12+x}{12+x}.

$$\frac{196x}{12+x}+\frac{-4x\left(12+x\right)}{12+x}=48$$

Since \frac{196x}{12+x} and \frac{-4x\left(12+x\right)}{12+x} have the same denominator, add them by adding their numerators.

$$\frac{196x-4x\left(12+x\right)}{12+x}=48$$

Do the multiplications in 196x-4x\left(12+x\right).

$$\frac{196x-48x-4x^{2}}{12+x}=48$$

Combine like terms in 196x-48x-4x^{2}.

$$\frac{148x-4x^{2}}{12+x}=48$$

Subtract 48 from both sides.

$$\frac{148x-4x^{2}}{12+x}-48=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 48 times \frac{12+x}{12+x}.

$$\frac{148x-4x^{2}}{12+x}-\frac{48\left(12+x\right)}{12+x}=0$$

Since \frac{148x-4x^{2}}{12+x} and \frac{48\left(12+x\right)}{12+x} have the same denominator, subtract them by subtracting their numerators.

$$\frac{148x-4x^{2}-48\left(12+x\right)}{12+x}=0$$

Do the multiplications in 148x-4x^{2}-48\left(12+x\right).

$$\frac{148x-4x^{2}-576-48x}{12+x}=0$$

Combine like terms in 148x-4x^{2}-576-48x.

$$\frac{100x-4x^{2}-576}{12+x}=0$$

Variable x cannot be equal to -12 since division by zero is not defined. Multiply both sides of the equation by x+12.

$$100x-4x^{2}-576=0$$

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.

$$-4x^{2}+100x-576=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 100 for b, and -576 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-4\right)\left(-576\right)}}{2\left(-4\right)}$$

Square 100.

$$x=\frac{-100±\sqrt{10000-4\left(-4\right)\left(-576\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-100±\sqrt{10000+16\left(-576\right)}}{2\left(-4\right)}$$

Multiply 16 times -576.

$$x=\frac{-100±\sqrt{10000-9216}}{2\left(-4\right)}$$

Add 10000 to -9216.

$$x=\frac{-100±\sqrt{784}}{2\left(-4\right)}$$

Take the square root of 784.

$$x=\frac{-100±28}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-100±28}{-8}$$

Now solve the equation x=\frac{-100±28}{-8} when ± is plus. Add -100 to 28.

$$x=-\frac{72}{-8}$$

Divide -72 by -8.

$$x=9$$

Now solve the equation x=\frac{-100±28}{-8} when ± is minus. Subtract 28 from -100.

$$x=-\frac{128}{-8}$$

Divide -128 by -8.

$$x=16$$

The equation is now solved.

$$x=9 x=16$$

Steps for Completing the Square

Variable x cannot be equal to -12 since division by zero is not defined. Multiply both sides of the equation by x+12.

$$14x\times \frac{14}{12+x}=4\left(x+12\right)$$

Express 14\times \frac{14}{12+x} as a single fraction.

$$\frac{14\times 14}{12+x}x=4\left(x+12\right)$$

Use the distributive property to multiply 4 by x+12.

$$\frac{14\times 14}{12+x}x=4x+48$$

Multiply 14 and 14 to get 196.

$$\frac{196}{12+x}x=4x+48$$

Express \frac{196}{12+x}x as a single fraction.

$$\frac{196x}{12+x}=4x+48$$

Subtract 4x from both sides.

$$\frac{196x}{12+x}-4x=48$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -4x times \frac{12+x}{12+x}.

$$\frac{196x}{12+x}+\frac{-4x\left(12+x\right)}{12+x}=48$$

Since \frac{196x}{12+x} and \frac{-4x\left(12+x\right)}{12+x} have the same denominator, add them by adding their numerators.

$$\frac{196x-4x\left(12+x\right)}{12+x}=48$$

Do the multiplications in 196x-4x\left(12+x\right).

$$\frac{196x-48x-4x^{2}}{12+x}=48$$

Combine like terms in 196x-48x-4x^{2}.

$$\frac{148x-4x^{2}}{12+x}=48$$

Variable x cannot be equal to -12 since division by zero is not defined. Multiply both sides of the equation by x+12.

$$148x-4x^{2}=48\left(x+12\right)$$

Use the distributive property to multiply 48 by x+12.

$$148x-4x^{2}=48x+576$$

Subtract 48x from both sides.

$$148x-4x^{2}-48x=576$$

Combine 148x and -48x to get 100x.

$$100x-4x^{2}=576$$

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.

$$-4x^{2}+100x=576$$

Divide both sides by -4.

$$\frac{-4x^{2}+100x}{-4}=\frac{576}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{100}{-4}x=\frac{576}{-4}$$

Divide 100 by -4.

$$x^{2}-25x=\frac{576}{-4}$$

Divide 576 by -4.

$$x^{2}-25x=-144$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=-144+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-25x+\frac{625}{4}=-144+\frac{625}{4}$$

Add -144 to \frac{625}{4}.

$$x^{2}-25x+\frac{625}{4}=\frac{49}{4}$$

Factor x^{2}-25x+\frac{625}{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}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x-\frac{25}{2}=\frac{7}{2} x-\frac{25}{2}=-\frac{7}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$x=16 x=9$$