Solve left(x+0right)left(35-2xright)=200

$$\left( x+0 \right) \left( 35-2x \right) = 200$$

Solve for x (complex solution)

$$x=\frac{-5\sqrt{15}i+35}{4}\approx 8.75-4.841229183i$$

$$x=\frac{35+5\sqrt{15}i}{4}\approx 8.75+4.841229183i$$

Steps Using the Quadratic Formula

Anything plus zero gives itself.

$$x\left(35-2x\right)=200$$

Use the distributive property to multiply x by 35-2x.

$$35x-2x^{2}=200$$

Subtract 200 from both sides.

$$35x-2x^{2}-200=0$$

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.

$$-2x^{2}+35x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 35 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-35±\sqrt{35^{2}-4\left(-2\right)\left(-200\right)}}{2\left(-2\right)}$$

Square 35.

$$x=\frac{-35±\sqrt{1225-4\left(-2\right)\left(-200\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-35±\sqrt{1225+8\left(-200\right)}}{2\left(-2\right)}$$

Multiply 8 times -200.

$$x=\frac{-35±\sqrt{1225-1600}}{2\left(-2\right)}$$

Add 1225 to -1600.

$$x=\frac{-35±\sqrt{-375}}{2\left(-2\right)}$$

Take the square root of -375.

$$x=\frac{-35±5\sqrt{15}i}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-35±5\sqrt{15}i}{-4}$$

Now solve the equation x=\frac{-35±5\sqrt{15}i}{-4} when ± is plus. Add -35 to 5i\sqrt{15}.

$$x=\frac{-35+5\sqrt{15}i}{-4}$$

Divide -35+5i\sqrt{15} by -4.

$$x=\frac{-5\sqrt{15}i+35}{4}$$

Now solve the equation x=\frac{-35±5\sqrt{15}i}{-4} when ± is minus. Subtract 5i\sqrt{15} from -35.

$$x=\frac{-5\sqrt{15}i-35}{-4}$$

Divide -35-5i\sqrt{15} by -4.

$$x=\frac{35+5\sqrt{15}i}{4}$$

The equation is now solved.

$$x=\frac{-5\sqrt{15}i+35}{4} x=\frac{35+5\sqrt{15}i}{4}$$

Steps for Completing the Square

Anything plus zero gives itself.

$$x\left(35-2x\right)=200$$

Use the distributive property to multiply x by 35-2x.

$$35x-2x^{2}=200$$

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.

$$-2x^{2}+35x=200$$

Divide both sides by -2.

$$\frac{-2x^{2}+35x}{-2}=\frac{200}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{35}{-2}x=\frac{200}{-2}$$

Divide 35 by -2.

$$x^{2}-\frac{35}{2}x=\frac{200}{-2}$$

Divide 200 by -2.

$$x^{2}-\frac{35}{2}x=-100$$

Divide -\frac{35}{2}, the coefficient of the x term, by 2 to get -\frac{35}{4}. Then add the square of -\frac{35}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{2}x+\left(-\frac{35}{4}\right)^{2}=-100+\left(-\frac{35}{4}\right)^{2}$$

Square -\frac{35}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{2}x+\frac{1225}{16}=-100+\frac{1225}{16}$$

Add -100 to \frac{1225}{16}.

$$x^{2}-\frac{35}{2}x+\frac{1225}{16}=-\frac{375}{16}$$

Factor x^{2}-\frac{35}{2}x+\frac{1225}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{4}\right)^{2}=-\frac{375}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{4}\right)^{2}}=\sqrt{-\frac{375}{16}}$$

Simplify.

$$x-\frac{35}{4}=\frac{5\sqrt{15}i}{4} x-\frac{35}{4}=-\frac{5\sqrt{15}i}{4}$$

Add \frac{35}{4} to both sides of the equation.

$$x=\frac{35+5\sqrt{15}i}{4} x=\frac{-5\sqrt{15}i+35}{4}$$