Solve (.150+x)(.250-x)/.0500=2.15

$$\frac { ( .150 + x ) ( .250 - x ) } { .0500 } = 2.15$$

Solve for x (complex solution)

$$x=\frac{1+i\times 3\sqrt{3}}{20}\approx 0.05+0.259807621i$$

$$x=\frac{-i\times 3\sqrt{3}+1}{20}\approx 0.05-0.259807621i$$

Steps Using the Quadratic Formula

Multiply both sides by 0.05.

$$\left(0.15+x\right)\left(0.25-x\right)=2.15\times 0.05$$

Use the distributive property to multiply 0.15+x by 0.25-x and combine like terms.

$$0.0375+0.1x-x^{2}=2.15\times 0.05$$

Multiply 2.15 and 0.05 to get 0.1075.

$$0.0375+0.1x-x^{2}=0.1075$$

Subtract 0.1075 from both sides.

$$0.0375+0.1x-x^{2}-0.1075=0$$

Subtract 0.1075 from 0.0375 to get -0.07.

$$-0.07+0.1x-x^{2}=0$$

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.

$$-x^{2}+0.1x-0.07=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0.1 for b, and -0.07 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.1±\sqrt{0.1^{2}-4\left(-1\right)\left(-0.07\right)}}{2\left(-1\right)}$$

Square 0.1 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.1±\sqrt{0.01-4\left(-1\right)\left(-0.07\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-0.1±\sqrt{0.01+4\left(-0.07\right)}}{2\left(-1\right)}$$

Multiply 4 times -0.07.

$$x=\frac{-0.1±\sqrt{0.01-0.28}}{2\left(-1\right)}$$

Add 0.01 to -0.28 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.1±\sqrt{-0.27}}{2\left(-1\right)}$$

Take the square root of -0.27.

$$x=\frac{-0.1±\frac{3\sqrt{3}i}{10}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-0.1±\frac{3\sqrt{3}i}{10}}{-2}$$

Now solve the equation x=\frac{-0.1±\frac{3\sqrt{3}i}{10}}{-2} when ± is plus. Add -0.1 to \frac{3i\sqrt{3}}{10}.

$$x=\frac{-1+3\sqrt{3}i}{-2\times 10}$$

Divide \frac{-1+3i\sqrt{3}}{10} by -2.

$$x=\frac{-3\sqrt{3}i+1}{20}$$

Now solve the equation x=\frac{-0.1±\frac{3\sqrt{3}i}{10}}{-2} when ± is minus. Subtract \frac{3i\sqrt{3}}{10} from -0.1.

$$x=\frac{-3\sqrt{3}i-1}{-2\times 10}$$

Divide \frac{-1-3i\sqrt{3}}{10} by -2.

$$x=\frac{1+3\sqrt{3}i}{20}$$

The equation is now solved.

$$x=\frac{-3\sqrt{3}i+1}{20} x=\frac{1+3\sqrt{3}i}{20}$$

Steps for Completing the Square

Multiply both sides by 0.05.

$$\left(0.15+x\right)\left(0.25-x\right)=2.15\times 0.05$$

Use the distributive property to multiply 0.15+x by 0.25-x and combine like terms.

$$0.0375+0.1x-x^{2}=2.15\times 0.05$$

Multiply 2.15 and 0.05 to get 0.1075.

$$0.0375+0.1x-x^{2}=0.1075$$

Subtract 0.0375 from both sides.

$$0.1x-x^{2}=0.1075-0.0375$$

Subtract 0.0375 from 0.1075 to get 0.07.

$$0.1x-x^{2}=0.07$$

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.

$$-x^{2}+0.1x=0.07$$

Divide both sides by -1.

$$\frac{-x^{2}+0.1x}{-1}=\frac{0.07}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{0.1}{-1}x=\frac{0.07}{-1}$$

Divide 0.1 by -1.

$$x^{2}-0.1x=\frac{0.07}{-1}$$

Divide 0.07 by -1.

$$x^{2}-0.1x=-0.07$$

Divide -0.1, the coefficient of the x term, by 2 to get -0.05. Then add the square of -0.05 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-0.1x+\left(-0.05\right)^{2}=-0.07+\left(-0.05\right)^{2}$$

Square -0.05 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-0.1x+0.0025=-0.07+0.0025$$

Add -0.07 to 0.0025 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-0.1x+0.0025=-0.0675$$

Factor x^{2}-0.1x+0.0025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-0.05\right)^{2}=-0.0675$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-0.05\right)^{2}}=\sqrt{-0.0675}$$

Simplify.

$$x-0.05=\frac{3\sqrt{3}i}{20} x-0.05=-\frac{3\sqrt{3}i}{20}$$

Add 0.05 to both sides of the equation.

$$x=\frac{1+3\sqrt{3}i}{20} x=\frac{-3\sqrt{3}i+1}{20}$$