Solve 40*76=x(34-x)

$$40 \times 76=x(34-x)$$

Solve for x (complex solution)

$$x=17+\sqrt{2751}i\approx 17+52.449976168i$$

$$x=-\sqrt{2751}i+17\approx 17-52.449976168i$$

Steps Using the Quadratic Formula

Multiply 40 and 76 to get 3040.

$$3040=x\left(34-x\right)$$

Use the distributive property to multiply x by 34-x.

$$3040=34x-x^{2}$$

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

$$34x-x^{2}=3040$$

Subtract 3040 from both sides.

$$34x-x^{2}-3040=0$$

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.

$$-x^{2}+34x-3040=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 34 for b, and -3040 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-1\right)\left(-3040\right)}}{2\left(-1\right)}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\left(-1\right)\left(-3040\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-34±\sqrt{1156+4\left(-3040\right)}}{2\left(-1\right)}$$

Multiply 4 times -3040.

$$x=\frac{-34±\sqrt{1156-12160}}{2\left(-1\right)}$$

Add 1156 to -12160.

$$x=\frac{-34±\sqrt{-11004}}{2\left(-1\right)}$$

Take the square root of -11004.

$$x=\frac{-34±2\sqrt{2751}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-34±2\sqrt{2751}i}{-2}$$

Now solve the equation x=\frac{-34±2\sqrt{2751}i}{-2} when ± is plus. Add -34 to 2i\sqrt{2751}.

$$x=\frac{-34+2\sqrt{2751}i}{-2}$$

Divide -34+2i\sqrt{2751} by -2.

$$x=-\sqrt{2751}i+17$$

Now solve the equation x=\frac{-34±2\sqrt{2751}i}{-2} when ± is minus. Subtract 2i\sqrt{2751} from -34.

$$x=\frac{-2\sqrt{2751}i-34}{-2}$$

Divide -34-2i\sqrt{2751} by -2.

$$x=17+\sqrt{2751}i$$

The equation is now solved.

$$x=-\sqrt{2751}i+17 x=17+\sqrt{2751}i$$

Steps for Completing the Square

Multiply 40 and 76 to get 3040.

$$3040=x\left(34-x\right)$$

Use the distributive property to multiply x by 34-x.

$$3040=34x-x^{2}$$

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

$$34x-x^{2}=3040$$

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.

$$-x^{2}+34x=3040$$

Divide both sides by -1.

$$\frac{-x^{2}+34x}{-1}=\frac{3040}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{34}{-1}x=\frac{3040}{-1}$$

Divide 34 by -1.

$$x^{2}-34x=\frac{3040}{-1}$$

Divide 3040 by -1.

$$x^{2}-34x=-3040$$

Divide -34, the coefficient of the x term, by 2 to get -17. Then add the square of -17 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-34x+\left(-17\right)^{2}=-3040+\left(-17\right)^{2}$$

Square -17.

$$x^{2}-34x+289=-3040+289$$

Add -3040 to 289.

$$x^{2}-34x+289=-2751$$

Factor x^{2}-34x+289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-17\right)^{2}=-2751$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-17\right)^{2}}=\sqrt{-2751}$$

Simplify.

$$x-17=\sqrt{2751}i x-17=-\sqrt{2751}i$$

Add 17 to both sides of the equation.

$$x=17+\sqrt{2751}i x=-\sqrt{2751}i+17$$