Solve 1320=3000-(10-2x)^2*30

$$1320 = 3000 - ( 10 - 2 x ) ^ { 2 } \cdot 30$$

Solve for x

$$x=5-\sqrt{14}\approx 1.258342613$$

$$x=\sqrt{14}+5\approx 8.741657387$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-2x\right)^{2}.

$$1320=3000-\left(100-40x+4x^{2}\right)\times 30$$

Use the distributive property to multiply 100-40x+4x^{2} by 30.

$$1320=3000-\left(3000-1200x+120x^{2}\right)$$

To find the opposite of 3000-1200x+120x^{2}, find the opposite of each term.

$$1320=3000-3000+1200x-120x^{2}$$

Subtract 3000 from 3000 to get 0.

$$1320=1200x-120x^{2}$$

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

$$1200x-120x^{2}=1320$$

Subtract 1320 from both sides.

$$1200x-120x^{2}-1320=0$$

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.

$$-120x^{2}+1200x-1320=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -120 for a, 1200 for b, and -1320 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1200±\sqrt{1200^{2}-4\left(-120\right)\left(-1320\right)}}{2\left(-120\right)}$$

Square 1200.

$$x=\frac{-1200±\sqrt{1440000-4\left(-120\right)\left(-1320\right)}}{2\left(-120\right)}$$

Multiply -4 times -120.

$$x=\frac{-1200±\sqrt{1440000+480\left(-1320\right)}}{2\left(-120\right)}$$

Multiply 480 times -1320.

$$x=\frac{-1200±\sqrt{1440000-633600}}{2\left(-120\right)}$$

Add 1440000 to -633600.

$$x=\frac{-1200±\sqrt{806400}}{2\left(-120\right)}$$

Take the square root of 806400.

$$x=\frac{-1200±240\sqrt{14}}{2\left(-120\right)}$$

Multiply 2 times -120.

$$x=\frac{-1200±240\sqrt{14}}{-240}$$

Now solve the equation x=\frac{-1200±240\sqrt{14}}{-240} when ± is plus. Add -1200 to 240\sqrt{14}.

$$x=\frac{240\sqrt{14}-1200}{-240}$$

Divide -1200+240\sqrt{14} by -240.

$$x=5-\sqrt{14}$$

Now solve the equation x=\frac{-1200±240\sqrt{14}}{-240} when ± is minus. Subtract 240\sqrt{14} from -1200.

$$x=\frac{-240\sqrt{14}-1200}{-240}$$

Divide -1200-240\sqrt{14} by -240.

$$x=\sqrt{14}+5$$

The equation is now solved.

$$x=5-\sqrt{14} x=\sqrt{14}+5$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-2x\right)^{2}.

$$1320=3000-\left(100-40x+4x^{2}\right)\times 30$$

Use the distributive property to multiply 100-40x+4x^{2} by 30.

$$1320=3000-\left(3000-1200x+120x^{2}\right)$$

To find the opposite of 3000-1200x+120x^{2}, find the opposite of each term.

$$1320=3000-3000+1200x-120x^{2}$$

Subtract 3000 from 3000 to get 0.

$$1320=1200x-120x^{2}$$

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

$$1200x-120x^{2}=1320$$

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.

$$-120x^{2}+1200x=1320$$

Divide both sides by -120.

$$\frac{-120x^{2}+1200x}{-120}=\frac{1320}{-120}$$

Dividing by -120 undoes the multiplication by -120.

$$x^{2}+\frac{1200}{-120}x=\frac{1320}{-120}$$

Divide 1200 by -120.

$$x^{2}-10x=\frac{1320}{-120}$$

Divide 1320 by -120.

$$x^{2}-10x=-11$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=-11+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=-11+25$$

Add -11 to 25.

$$x^{2}-10x+25=14$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{14}$$

Simplify.

$$x-5=\sqrt{14} x-5=-\sqrt{14}$$

Add 5 to both sides of the equation.

$$x=\sqrt{14}+5 x=5-\sqrt{14}$$