Solve (40*25)-(left(40-xright)left(25-xright))=1156

$$(40 \times 25)-( \left( 40-x \right) \left( 25-x \right) )=1156$$

Solve for x (complex solution)

$$x=\frac{65+\sqrt{399}i}{2}\approx 32.5+9.987492178i$$

$$x=\frac{-\sqrt{399}i+65}{2}\approx 32.5-9.987492178i$$

Steps Using the Quadratic Formula

Multiply 40 and 25 to get 1000.

$$1000-\left(40-x\right)\left(25-x\right)=1156$$

Use the distributive property to multiply 40-x by 25-x and combine like terms.

$$1000-\left(1000-65x+x^{2}\right)=1156$$

To find the opposite of 1000-65x+x^{2}, find the opposite of each term.

$$1000-1000+65x-x^{2}=1156$$

Subtract 1000 from 1000 to get 0.

$$65x-x^{2}=1156$$

Subtract 1156 from both sides.

$$65x-x^{2}-1156=0$$

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.

$$-x^{2}+65x-1156=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 65 for b, and -1156 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-65±\sqrt{65^{2}-4\left(-1\right)\left(-1156\right)}}{2\left(-1\right)}$$

Square 65.

$$x=\frac{-65±\sqrt{4225-4\left(-1\right)\left(-1156\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-65±\sqrt{4225+4\left(-1156\right)}}{2\left(-1\right)}$$

Multiply 4 times -1156.

$$x=\frac{-65±\sqrt{4225-4624}}{2\left(-1\right)}$$

Add 4225 to -4624.

$$x=\frac{-65±\sqrt{-399}}{2\left(-1\right)}$$

Take the square root of -399.

$$x=\frac{-65±\sqrt{399}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-65±\sqrt{399}i}{-2}$$

Now solve the equation x=\frac{-65±\sqrt{399}i}{-2} when ± is plus. Add -65 to i\sqrt{399}.

$$x=\frac{-65+\sqrt{399}i}{-2}$$

Divide -65+i\sqrt{399} by -2.

$$x=\frac{-\sqrt{399}i+65}{2}$$

Now solve the equation x=\frac{-65±\sqrt{399}i}{-2} when ± is minus. Subtract i\sqrt{399} from -65.

$$x=\frac{-\sqrt{399}i-65}{-2}$$

Divide -65-i\sqrt{399} by -2.

$$x=\frac{65+\sqrt{399}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{399}i+65}{2} x=\frac{65+\sqrt{399}i}{2}$$

Steps for Completing the Square

Multiply 40 and 25 to get 1000.

$$1000-\left(40-x\right)\left(25-x\right)=1156$$

Use the distributive property to multiply 40-x by 25-x and combine like terms.

$$1000-\left(1000-65x+x^{2}\right)=1156$$

To find the opposite of 1000-65x+x^{2}, find the opposite of each term.

$$1000-1000+65x-x^{2}=1156$$

Subtract 1000 from 1000 to get 0.

$$65x-x^{2}=1156$$

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.

$$-x^{2}+65x=1156$$

Divide both sides by -1.

$$\frac{-x^{2}+65x}{-1}=\frac{1156}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{65}{-1}x=\frac{1156}{-1}$$

Divide 65 by -1.

$$x^{2}-65x=\frac{1156}{-1}$$

Divide 1156 by -1.

$$x^{2}-65x=-1156$$

Divide -65, the coefficient of the x term, by 2 to get -\frac{65}{2}. Then add the square of -\frac{65}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-65x+\left(-\frac{65}{2}\right)^{2}=-1156+\left(-\frac{65}{2}\right)^{2}$$

Square -\frac{65}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-65x+\frac{4225}{4}=-1156+\frac{4225}{4}$$

Add -1156 to \frac{4225}{4}.

$$x^{2}-65x+\frac{4225}{4}=-\frac{399}{4}$$

Factor x^{2}-65x+\frac{4225}{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}.

$$\left(x-\frac{65}{2}\right)^{2}=-\frac{399}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{65}{2}\right)^{2}}=\sqrt{-\frac{399}{4}}$$

Simplify.

$$x-\frac{65}{2}=\frac{\sqrt{399}i}{2} x-\frac{65}{2}=-\frac{\sqrt{399}i}{2}$$

Add \frac{65}{2} to both sides of the equation.

$$x=\frac{65+\sqrt{399}i}{2} x=\frac{-\sqrt{399}i+65}{2}$$