Solve 0.5/x=4.5-x/10

$$\frac { 0.5 } { x } = \frac { 4.5 - x } { 10 }$$

Solve for x

$$x=2$$

$$x=2.5$$

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 x,10.

$$10\times 0.5=x\left(4.5-x\right)$$

Multiply 10 and 0.5 to get 5.

$$5=x\left(4.5-x\right)$$

Use the distributive property to multiply x by 4.5-x.

$$5=4.5x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$4.5x-x^{2}=5$$

Subtract 5 from both sides.

$$4.5x-x^{2}-5=0$$

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.

$$-x^{2}+4.5x-5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4.5 for b, and -5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4.5±\sqrt{4.5^{2}-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Square 4.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-4.5±\sqrt{20.25-4\left(-1\right)\left(-5\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4.5±\sqrt{20.25+4\left(-5\right)}}{2\left(-1\right)}$$

Multiply 4 times -5.

$$x=\frac{-4.5±\sqrt{20.25-20}}{2\left(-1\right)}$$

Add 20.25 to -20.

$$x=\frac{-4.5±\sqrt{0.25}}{2\left(-1\right)}$$

Take the square root of 0.25.

$$x=\frac{-4.5±\frac{1}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4.5±\frac{1}{2}}{-2}$$

Now solve the equation x=\frac{-4.5±\frac{1}{2}}{-2} when ± is plus. Add -4.5 to \frac{1}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{4}{-2}$$

Divide -4 by -2.

$$x=2$$

Now solve the equation x=\frac{-4.5±\frac{1}{2}}{-2} when ± is minus. Subtract \frac{1}{2} from -4.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{5}{-2}$$

Divide -5 by -2.

$$x=\frac{5}{2}$$

The equation is now solved.

$$x=2 x=\frac{5}{2}$$

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 x,10.

$$10\times 0.5=x\left(4.5-x\right)$$

Multiply 10 and 0.5 to get 5.

$$5=x\left(4.5-x\right)$$

Use the distributive property to multiply x by 4.5-x.

$$5=4.5x-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$4.5x-x^{2}=5$$

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.

$$-x^{2}+4.5x=5$$

Divide both sides by -1.

$$\frac{-x^{2}+4.5x}{-1}=\frac{5}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4.5}{-1}x=\frac{5}{-1}$$

Divide 4.5 by -1.

$$x^{2}-4.5x=\frac{5}{-1}$$

Divide 5 by -1.

$$x^{2}-4.5x=-5$$

Divide -4.5, the coefficient of the x term, by 2 to get -2.25. Then add the square of -2.25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4.5x+\left(-2.25\right)^{2}=-5+\left(-2.25\right)^{2}$$

Square -2.25 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-4.5x+5.0625=-5+5.0625$$

Add -5 to 5.0625.

$$x^{2}-4.5x+5.0625=0.0625$$

Factor x^{2}-4.5x+5.0625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2.25\right)^{2}=0.0625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2.25\right)^{2}}=\sqrt{0.0625}$$

Simplify.

$$x-2.25=\frac{1}{4} x-2.25=-\frac{1}{4}$$

Add 2.25 to both sides of the equation.

$$x=\frac{5}{2} x=2$$