Solve x(65-x)+3x-2(65-x)-6=1089

$$x ( 65 - x ) + 3 x - 2 ( 65 - x ) - 6 = 1089$$

Solve for x

$$x=35$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 65-x.

$$65x-x^{2}+3x-2\left(65-x\right)-6=1089$$

Combine 65x and 3x to get 68x.

$$68x-x^{2}-2\left(65-x\right)-6=1089$$

Use the distributive property to multiply -2 by 65-x.

$$68x-x^{2}-130+2x-6=1089$$

Combine 68x and 2x to get 70x.

$$70x-x^{2}-130-6=1089$$

Subtract 6 from -130 to get -136.

$$70x-x^{2}-136=1089$$

Subtract 1089 from both sides.

$$70x-x^{2}-136-1089=0$$

Subtract 1089 from -136 to get -1225.

$$70x-x^{2}-1225=0$$

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.

$$-x^{2}+70x-1225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 70 for b, and -1225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-70±\sqrt{70^{2}-4\left(-1\right)\left(-1225\right)}}{2\left(-1\right)}$$

Square 70.

$$x=\frac{-70±\sqrt{4900-4\left(-1\right)\left(-1225\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-70±\sqrt{4900+4\left(-1225\right)}}{2\left(-1\right)}$$

Multiply 4 times -1225.

$$x=\frac{-70±\sqrt{4900-4900}}{2\left(-1\right)}$$

Add 4900 to -4900.

$$x=\frac{-70±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=-\frac{70}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=-\frac{70}{-2}$$

Divide -70 by -2.

$$x=35$$

Steps for Completing the Square

Use the distributive property to multiply x by 65-x.

$$65x-x^{2}+3x-2\left(65-x\right)-6=1089$$

Combine 65x and 3x to get 68x.

$$68x-x^{2}-2\left(65-x\right)-6=1089$$

Use the distributive property to multiply -2 by 65-x.

$$68x-x^{2}-130+2x-6=1089$$

Combine 68x and 2x to get 70x.

$$70x-x^{2}-130-6=1089$$

Subtract 6 from -130 to get -136.

$$70x-x^{2}-136=1089$$

Add 136 to both sides.

$$70x-x^{2}=1089+136$$

Add 1089 and 136 to get 1225.

$$70x-x^{2}=1225$$

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.

$$-x^{2}+70x=1225$$

Divide both sides by -1.

$$\frac{-x^{2}+70x}{-1}=\frac{1225}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{70}{-1}x=\frac{1225}{-1}$$

Divide 70 by -1.

$$x^{2}-70x=\frac{1225}{-1}$$

Divide 1225 by -1.

$$x^{2}-70x=-1225$$

Divide -70, the coefficient of the x term, by 2 to get -35. Then add the square of -35 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-70x+\left(-35\right)^{2}=-1225+\left(-35\right)^{2}$$

Square -35.

$$x^{2}-70x+1225=-1225+1225$$

Add -1225 to 1225.

$$x^{2}-70x+1225=0$$

Factor x^{2}-70x+1225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-35\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-35\right)^{2}}=\sqrt{0}$$

Simplify.

$$x-35=0 x-35=0$$

Add 35 to both sides of the equation.

$$x=35 x=35$$

The equation is now solved. Solutions are the same.

$$x=35$$