Solve 35*15=(19-x)(15+x)

$$35 \times 15=(19-x)(15+x)$$

Solve for x (complex solution)

$$x=2+2\sqrt{59}i\approx 2+15.362291496i$$

$$x=-2\sqrt{59}i+2\approx 2-15.362291496i$$

Steps Using the Quadratic Formula

Multiply 35 and 15 to get 525.

$$525=\left(19-x\right)\left(15+x\right)$$

Use the distributive property to multiply 19-x by 15+x and combine like terms.

$$525=285+4x-x^{2}$$

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

$$285+4x-x^{2}=525$$

Subtract 525 from both sides.

$$285+4x-x^{2}-525=0$$

Subtract 525 from 285 to get -240.

$$-240+4x-x^{2}=0$$

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.

$$-x^{2}+4x-240=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and -240 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\left(-240\right)}}{2\left(-1\right)}$$

Multiply 4 times -240.

$$x=\frac{-4±\sqrt{16-960}}{2\left(-1\right)}$$

Add 16 to -960.

$$x=\frac{-4±\sqrt{-944}}{2\left(-1\right)}$$

Take the square root of -944.

$$x=\frac{-4±4\sqrt{59}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±4\sqrt{59}i}{-2}$$

Now solve the equation x=\frac{-4±4\sqrt{59}i}{-2} when ± is plus. Add -4 to 4i\sqrt{59}.

$$x=\frac{-4+4\sqrt{59}i}{-2}$$

Divide -4+4i\sqrt{59} by -2.

$$x=-2\sqrt{59}i+2$$

Now solve the equation x=\frac{-4±4\sqrt{59}i}{-2} when ± is minus. Subtract 4i\sqrt{59} from -4.

$$x=\frac{-4\sqrt{59}i-4}{-2}$$

Divide -4-4i\sqrt{59} by -2.

$$x=2+2\sqrt{59}i$$

The equation is now solved.

$$x=-2\sqrt{59}i+2 x=2+2\sqrt{59}i$$

Steps for Completing the Square

Multiply 35 and 15 to get 525.

$$525=\left(19-x\right)\left(15+x\right)$$

Use the distributive property to multiply 19-x by 15+x and combine like terms.

$$525=285+4x-x^{2}$$

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

$$285+4x-x^{2}=525$$

Subtract 285 from both sides.

$$4x-x^{2}=525-285$$

Subtract 285 from 525 to get 240.

$$4x-x^{2}=240$$

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.

$$-x^{2}+4x=240$$

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=\frac{240}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=\frac{240}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=\frac{240}{-1}$$

Divide 240 by -1.

$$x^{2}-4x=-240$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-240+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=-240+4$$

Add -240 to 4.

$$x^{2}-4x+4=-236$$

Factor x^{2}-4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-2\right)^{2}=-236$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-236}$$

Simplify.

$$x-2=2\sqrt{59}i x-2=-2\sqrt{59}i$$

Add 2 to both sides of the equation.

$$x=2+2\sqrt{59}i x=-2\sqrt{59}i+2$$