Solve (-0.5x+80)(70+x)=6759

$$( - 0.5 x + 80 ) ( 70 + x ) = 6759$$

Solve for x (complex solution)

$$x=45+i\sqrt{293}\approx 45+17.117242769i$$

$$x=-i\sqrt{293}+45\approx 45-17.117242769i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -0.5x+80 by 70+x and combine like terms.

$$45x-0.5x^{2}+5600=6759$$

Subtract 6759 from both sides.

$$45x-0.5x^{2}[phone]=0$$

Subtract 6759 from 5600 to get -1159.

$$45x-0.5x^{2}-1159=0$$

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.

$$-0.5x^{2}+45x-1159=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.5 for a, 45 for b, and -1159 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45±\sqrt{45^{2}-4\left(-0.5\right)\left(-1159\right)}}{2\left(-0.5\right)}$$

Square 45.

$$x=\frac{-45±\sqrt{2025-4\left(-0.5\right)\left(-1159\right)}}{2\left(-0.5\right)}$$

Multiply -4 times -0.5.

$$x=\frac{-45±\sqrt{2025+2\left(-1159\right)}}{2\left(-0.5\right)}$$

Multiply 2 times -1159.

$$x=\frac{-45±\sqrt{2025-2318}}{2\left(-0.5\right)}$$

Add 2025 to -2318.

$$x=\frac{-45±\sqrt{-293}}{2\left(-0.5\right)}$$

Take the square root of -293.

$$x=\frac{-45±\sqrt{293}i}{2\left(-0.5\right)}$$

Multiply 2 times -0.5.

$$x=\frac{-45±\sqrt{293}i}{-1}$$

Now solve the equation x=\frac{-45±\sqrt{293}i}{-1} when ± is plus. Add -45 to i\sqrt{293}.

$$x=\frac{-45+\sqrt{293}i}{-1}$$

Divide -45+i\sqrt{293} by -1.

$$x=-\sqrt{293}i+45$$

Now solve the equation x=\frac{-45±\sqrt{293}i}{-1} when ± is minus. Subtract i\sqrt{293} from -45.

$$x=\frac{-\sqrt{293}i-45}{-1}$$

Divide -45-i\sqrt{293} by -1.

$$x=45+\sqrt{293}i$$

The equation is now solved.

$$x=-\sqrt{293}i+45 x=45+\sqrt{293}i$$

Steps for Completing the Square

Use the distributive property to multiply -0.5x+80 by 70+x and combine like terms.

$$45x-0.5x^{2}+5600=6759$$

Subtract 5600 from both sides.

$$45x-0.5x^{2}=6759-5600$$

Subtract 5600 from 6759 to get 1159.

$$45x-0.5x^{2}=1159$$

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.

$$-0.5x^{2}+45x=1159$$

Multiply both sides by -2.

$$\frac{-0.5x^{2}+45x}{-0.5}=\frac{1159}{-0.5}$$

Dividing by -0.5 undoes the multiplication by -0.5.

$$x^{2}+\frac{45}{-0.5}x=\frac{1159}{-0.5}$$

Divide 45 by -0.5 by multiplying 45 by the reciprocal of -0.5.

$$x^{2}-90x=\frac{1159}{-0.5}$$

Divide 1159 by -0.5 by multiplying 1159 by the reciprocal of -0.5.

$$x^{2}-90x=-2318$$

Divide -90, the coefficient of the x term, by 2 to get -45. Then add the square of -45 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-90x+\left(-45\right)^{2}=-2318+\left(-45\right)^{2}$$

Square -45.

$$x^{2}-90x+2025=-2318+2025$$

Add -2318 to 2025.

$$x^{2}-90x+2025=-293$$

Factor x^{2}-90x+2025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-45\right)^{2}=-293$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-45\right)^{2}}=\sqrt{-293}$$

Simplify.

$$x-45=\sqrt{293}i x-45=-\sqrt{293}i$$

Add 45 to both sides of the equation.

$$x=45+\sqrt{293}i x=-\sqrt{293}i+45$$