Solve 16+4/5x=9/10x

$$16 + \frac { 4 } { 5 } x = \frac { 9 } { 10 x }$$

Solve for x

$$x=\frac{\sqrt{1618}}{4}-10\approx 0.056092681$$

$$x=-\frac{\sqrt{1618}}{4}-10\approx -20.056092681$$

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 10x, the least common multiple of 5,10x.

$$10x\times 16+\frac{4}{5}x\times 10x=9$$

Multiply 10 and 16 to get 160.

$$160x+\frac{4}{5}x\times 10x=9$$

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

$$160x+\frac{4}{5}x^{2}\times 10=9$$

Multiply \frac{4}{5} and 10 to get 8.

$$160x+8x^{2}=9$$

Subtract 9 from both sides.

$$160x+8x^{2}-9=0$$

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.

$$8x^{2}+160x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 160 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-160±\sqrt{160^{2}-4\times 8\left(-9\right)}}{2\times 8}$$

Square 160.

$$x=\frac{-160±\sqrt{25600-4\times 8\left(-9\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-160±\sqrt{25600-32\left(-9\right)}}{2\times 8}$$

Multiply -32 times -9.

$$x=\frac{-160±\sqrt{25600+288}}{2\times 8}$$

Add 25600 to 288.

$$x=\frac{-160±\sqrt{25888}}{2\times 8}$$

Take the square root of 25888.

$$x=\frac{-160±4\sqrt{1618}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-160±4\sqrt{1618}}{16}$$

Now solve the equation x=\frac{-160±4\sqrt{1618}}{16} when ± is plus. Add -160 to 4\sqrt{1618}.

$$x=\frac{4\sqrt{1618}-160}{16}$$

Divide -160+4\sqrt{1618} by 16.

$$x=\frac{\sqrt{1618}}{4}-10$$

Now solve the equation x=\frac{-160±4\sqrt{1618}}{16} when ± is minus. Subtract 4\sqrt{1618} from -160.

$$x=\frac{-4\sqrt{1618}-160}{16}$$

Divide -160-4\sqrt{1618} by 16.

$$x=-\frac{\sqrt{1618}}{4}-10$$

The equation is now solved.

$$x=\frac{\sqrt{1618}}{4}-10 x=-\frac{\sqrt{1618}}{4}-10$$

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 10x, the least common multiple of 5,10x.

$$10x\times 16+\frac{4}{5}x\times 10x=9$$

Multiply 10 and 16 to get 160.

$$160x+\frac{4}{5}x\times 10x=9$$

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

$$160x+\frac{4}{5}x^{2}\times 10=9$$

Multiply \frac{4}{5} and 10 to get 8.

$$160x+8x^{2}=9$$

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.

$$8x^{2}+160x=9$$

Divide both sides by 8.

$$\frac{8x^{2}+160x}{8}=\frac{9}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{160}{8}x=\frac{9}{8}$$

Divide 160 by 8.

$$x^{2}+20x=\frac{9}{8}$$

Divide 20, the coefficient of the x term, by 2 to get 10. Then add the square of 10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+20x+10^{2}=\frac{9}{8}+10^{2}$$

Square 10.

$$x^{2}+20x+100=\frac{9}{8}+100$$

Add \frac{9}{8} to 100.

$$x^{2}+20x+100=\frac{809}{8}$$

Factor x^{2}+20x+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+10\right)^{2}=\frac{809}{8}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{\frac{809}{8}}$$

Simplify.

$$x+10=\frac{\sqrt{1618}}{4} x+10=-\frac{\sqrt{1618}}{4}$$

Subtract 10 from both sides of the equation.

$$x=\frac{\sqrt{1618}}{4}-10 x=-\frac{\sqrt{1618}}{4}-10$$