Solve 78/x+8*7=2x

$$7 \frac{ 8 }{ x } +8 \cdot 7 = 2x$$

Solve for x

$$x=4\sqrt{14}+14\approx 28.966629547$$

$$x=14-4\sqrt{14}\approx -0.966629547$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$7\times 8+8\times 7x=2xx$$

Multiply x and x to get x^{2}.

$$7\times 8+8\times 7x=2x^{2}$$

Multiply 7 and 8 to get 56. Multiply 8 and 7 to get 56.

$$56+56x=2x^{2}$$

Subtract 2x^{2} from both sides.

$$56+56x-2x^{2}=0$$

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.

$$-2x^{2}+56x+56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 56 for b, and 56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-56±\sqrt{56^{2}-4\left(-2\right)\times 56}}{2\left(-2\right)}$$

Square 56.

$$x=\frac{-56±\sqrt{3136-4\left(-2\right)\times 56}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-56±\sqrt{3136+8\times 56}}{2\left(-2\right)}$$

Multiply 8 times 56.

$$x=\frac{-56±\sqrt{3136+448}}{2\left(-2\right)}$$

Add 3136 to 448.

$$x=\frac{-56±\sqrt{3584}}{2\left(-2\right)}$$

Take the square root of 3584.

$$x=\frac{-56±16\sqrt{14}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-56±16\sqrt{14}}{-4}$$

Now solve the equation x=\frac{-56±16\sqrt{14}}{-4} when ± is plus. Add -56 to 16\sqrt{14}.

$$x=\frac{16\sqrt{14}-56}{-4}$$

Divide -56+16\sqrt{14} by -4.

$$x=14-4\sqrt{14}$$

Now solve the equation x=\frac{-56±16\sqrt{14}}{-4} when ± is minus. Subtract 16\sqrt{14} from -56.

$$x=\frac{-16\sqrt{14}-56}{-4}$$

Divide -56-16\sqrt{14} by -4.

$$x=4\sqrt{14}+14$$

The equation is now solved.

$$x=14-4\sqrt{14} x=4\sqrt{14}+14$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$7\times 8+8\times 7x=2xx$$

Multiply x and x to get x^{2}.

$$7\times 8+8\times 7x=2x^{2}$$

Multiply 7 and 8 to get 56. Multiply 8 and 7 to get 56.

$$56+56x=2x^{2}$$

Subtract 2x^{2} from both sides.

$$56+56x-2x^{2}=0$$

Subtract 56 from both sides. Anything subtracted from zero gives its negation.

$$56x-2x^{2}=-56$$

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.

$$-2x^{2}+56x=-56$$

Divide both sides by -2.

$$\frac{-2x^{2}+56x}{-2}=-\frac{56}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{56}{-2}x=-\frac{56}{-2}$$

Divide 56 by -2.

$$x^{2}-28x=-\frac{56}{-2}$$

Divide -56 by -2.

$$x^{2}-28x=28$$

Divide -28, the coefficient of the x term, by 2 to get -14. Then add the square of -14 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-28x+\left(-14\right)^{2}=28+\left(-14\right)^{2}$$

Square -14.

$$x^{2}-28x+196=28+196$$

Add 28 to 196.

$$x^{2}-28x+196=224$$

Factor x^{2}-28x+196. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-14\right)^{2}=224$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-14\right)^{2}}=\sqrt{224}$$

Simplify.

$$x-14=4\sqrt{14} x-14=-4\sqrt{14}$$

Add 14 to both sides of the equation.

$$x=4\sqrt{14}+14 x=14-4\sqrt{14}$$