Solve 32x+2*20x-2x^2=32*20-566

$$32 x + 2 \times 20 x - 2 x ^ { 2 } = 32 \times 20 - 566$$

Solve for x

$$x=\sqrt{287}+18\approx 34.941074346$$

$$x=18-\sqrt{287}\approx 1.058925654$$

Steps Using the Quadratic Formula

Multiply 2 and 20 to get 40.

$$32x+40x-2x^{2}=32\times 20-566$$

Combine 32x and 40x to get 72x.

$$72x-2x^{2}=32\times 20-566$$

Multiply 32 and 20 to get 640.

$$72x-2x^{2}=640-566$$

Subtract 566 from 640 to get 74.

$$72x-2x^{2}=74$$

Subtract 74 from both sides.

$$72x-2x^{2}-74=0$$

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.

$$-2x^{2}+72x-74=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 72 for b, and -74 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-72±\sqrt{72^{2}-4\left(-2\right)\left(-74\right)}}{2\left(-2\right)}$$

Square 72.

$$x=\frac{-72±\sqrt{5184-4\left(-2\right)\left(-74\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-72±\sqrt{5184+8\left(-74\right)}}{2\left(-2\right)}$$

Multiply 8 times -74.

$$x=\frac{-72±\sqrt{5184-592}}{2\left(-2\right)}$$

Add 5184 to -592.

$$x=\frac{-72±\sqrt{4592}}{2\left(-2\right)}$$

Take the square root of 4592.

$$x=\frac{-72±4\sqrt{287}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-72±4\sqrt{287}}{-4}$$

Now solve the equation x=\frac{-72±4\sqrt{287}}{-4} when ± is plus. Add -72 to 4\sqrt{287}.

$$x=\frac{4\sqrt{287}-72}{-4}$$

Divide -72+4\sqrt{287} by -4.

$$x=18-\sqrt{287}$$

Now solve the equation x=\frac{-72±4\sqrt{287}}{-4} when ± is minus. Subtract 4\sqrt{287} from -72.

$$x=\frac{-4\sqrt{287}-72}{-4}$$

Divide -72-4\sqrt{287} by -4.

$$x=\sqrt{287}+18$$

The equation is now solved.

$$x=18-\sqrt{287} x=\sqrt{287}+18$$

Steps for Completing the Square

Multiply 2 and 20 to get 40.

$$32x+40x-2x^{2}=32\times 20-566$$

Combine 32x and 40x to get 72x.

$$72x-2x^{2}=32\times 20-566$$

Multiply 32 and 20 to get 640.

$$72x-2x^{2}=640-566$$

Subtract 566 from 640 to get 74.

$$72x-2x^{2}=74$$

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.

$$-2x^{2}+72x=74$$

Divide both sides by -2.

$$\frac{-2x^{2}+72x}{-2}=\frac{74}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{72}{-2}x=\frac{74}{-2}$$

Divide 72 by -2.

$$x^{2}-36x=\frac{74}{-2}$$

Divide 74 by -2.

$$x^{2}-36x=-37$$

Divide -36, the coefficient of the x term, by 2 to get -18. Then add the square of -18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-36x+\left(-18\right)^{2}=-37+\left(-18\right)^{2}$$

Square -18.

$$x^{2}-36x+324=-37+324$$

Add -37 to 324.

$$x^{2}-36x+324=287$$

Factor x^{2}-36x+324. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-18\right)^{2}=287$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-18\right)^{2}}=\sqrt{287}$$

Simplify.

$$x-18=\sqrt{287} x-18=-\sqrt{287}$$

Add 18 to both sides of the equation.

$$x=\sqrt{287}+18 x=18-\sqrt{287}$$