Solve 4x/x-8=12/x-7

$$\frac{ 4x }{ x-8 } = \frac{ 12 }{ x-7 }$$

Solve for x

$$x=4$$

$$x=6$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 7,8 since division by zero is not defined. Multiply both sides of the equation by \left(x-8\right)\left(x-7\right), the least common multiple of x-8,x-7.

$$\left(x-7\right)\times 4x=\left(x-8\right)\times 12$$

Use the distributive property to multiply x-7 by 4.

$$\left(4x-28\right)x=\left(x-8\right)\times 12$$

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

$$4x^{2}-28x=\left(x-8\right)\times 12$$

Use the distributive property to multiply x-8 by 12.

$$4x^{2}-28x=12x-96$$

Subtract 12x from both sides.

$$4x^{2}-28x-12x=-96$$

Combine -28x and -12x to get -40x.

$$4x^{2}-40x=-96$$

Add 96 to both sides.

$$4x^{2}-40x+96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -40 for b, and 96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 4\times 96}}{2\times 4}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 4\times 96}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-40\right)±\sqrt{1600-16\times 96}}{2\times 4}$$

Multiply -16 times 96.

$$x=\frac{-\left(-40\right)±\sqrt{1600-1536}}{2\times 4}$$

Add 1600 to -1536.

$$x=\frac{-\left(-40\right)±\sqrt{64}}{2\times 4}$$

Take the square root of 64.

$$x=\frac{-\left(-40\right)±8}{2\times 4}$$

The opposite of -40 is 40.

$$x=\frac{40±8}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{40±8}{8}$$

Now solve the equation x=\frac{40±8}{8} when ± is plus. Add 40 to 8.

$$x=\frac{48}{8}$$

Divide 48 by 8.

$$x=6$$

Now solve the equation x=\frac{40±8}{8} when ± is minus. Subtract 8 from 40.

$$x=\frac{32}{8}$$

Divide 32 by 8.

$$x=4$$

The equation is now solved.

$$x=6 x=4$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 7,8 since division by zero is not defined. Multiply both sides of the equation by \left(x-8\right)\left(x-7\right), the least common multiple of x-8,x-7.

$$\left(x-7\right)\times 4x=\left(x-8\right)\times 12$$

Use the distributive property to multiply x-7 by 4.

$$\left(4x-28\right)x=\left(x-8\right)\times 12$$

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

$$4x^{2}-28x=\left(x-8\right)\times 12$$

Use the distributive property to multiply x-8 by 12.

$$4x^{2}-28x=12x-96$$

Subtract 12x from both sides.

$$4x^{2}-28x-12x=-96$$

Combine -28x and -12x to get -40x.

$$4x^{2}-40x=-96$$

Divide both sides by 4.

$$\frac{4x^{2}-40x}{4}=-\frac{96}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{40}{4}\right)x=-\frac{96}{4}$$

Divide -40 by 4.

$$x^{2}-10x=-\frac{96}{4}$$

Divide -96 by 4.

$$x^{2}-10x=-24$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-24+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-24+25$$

Add -24 to 25.

$$x^{2}-10x+25=1$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{1}$$

Simplify.

$$x-5=1 x-5=-1$$

Add 5 to both sides of the equation.

$$x=6 x=4$$