Solve 4420=-5(x-70)^2+4500

$$4420 = - 5 ( x - 70 ) ^ { 2 } + 4500$$

Solve for x

$$x=74$$

$$x=66$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-70\right)^{2}.

$$4420=-5\left(x^{2}-140x+4900\right)+4500$$

Use the distributive property to multiply -5 by x^{2}-140x+4900.

$$4420=-5x^{2}+700x-24500+4500$$

Add -24500 and 4500 to get -20000.

$$4420=-5x^{2}+700x-20000$$

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

$$-5x^{2}+700x-20000=4420$$

Subtract 4420 from both sides.

$$-5x^{2}+700x-20000-4420=0$$

Subtract 4420 from -20000 to get -24420.

$$-5x^{2}+700x-24420=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 700 for b, and -24420 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-700±\sqrt{700^{2}-4\left(-5\right)\left(-24420\right)}}{2\left(-5\right)}$$

Square 700.

$$x=\frac{-700±\sqrt{490000-4\left(-5\right)\left(-24420\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-700±\sqrt{490000+20\left(-24420\right)}}{2\left(-5\right)}$$

Multiply 20 times -24420.

$$x=\frac{-700±\sqrt{490000-488400}}{2\left(-5\right)}$$

Add 490000 to -488400.

$$x=\frac{-700±\sqrt{1600}}{2\left(-5\right)}$$

Take the square root of 1600.

$$x=\frac{-700±40}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-700±40}{-10}$$

Now solve the equation x=\frac{-700±40}{-10} when ± is plus. Add -700 to 40.

$$x=-\frac{660}{-10}$$

Divide -660 by -10.

$$x=66$$

Now solve the equation x=\frac{-700±40}{-10} when ± is minus. Subtract 40 from -700.

$$x=-\frac{740}{-10}$$

Divide -740 by -10.

$$x=74$$

The equation is now solved.

$$x=66 x=74$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-70\right)^{2}.

$$4420=-5\left(x^{2}-140x+4900\right)+4500$$

Use the distributive property to multiply -5 by x^{2}-140x+4900.

$$4420=-5x^{2}+700x-24500+4500$$

Add -24500 and 4500 to get -20000.

$$4420=-5x^{2}+700x-20000$$

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

$$-5x^{2}+700x-20000=4420$$

Add 20000 to both sides.

$$-5x^{2}+700x=4420+20000$$

Add 4420 and 20000 to get 24420.

$$-5x^{2}+700x=24420$$

Divide both sides by -5.

$$\frac{-5x^{2}+700x}{-5}=\frac{24420}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{700}{-5}x=\frac{24420}{-5}$$

Divide 700 by -5.

$$x^{2}-140x=\frac{24420}{-5}$$

Divide 24420 by -5.

$$x^{2}-140x=-4884$$

Divide -140, the coefficient of the x term, by 2 to get -70. Then add the square of -70 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-140x+\left(-70\right)^{2}=-4884+\left(-70\right)^{2}$$

Square -70.

$$x^{2}-140x+4900=-4884+4900$$

Add -4884 to 4900.

$$x^{2}-140x+4900=16$$

Factor x^{2}-140x+4900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-70\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-70\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-70=4 x-70=-4$$

Add 70 to both sides of the equation.

$$x=74 x=66$$