Solve {l}{120,400=6,400x-18x^2-400}{text{Solvetheequation.(Enteryouranswers}}{x=}

$$\left. \begin{array} { l } { 120,400 = 6,400 x - 18 x ^ { 2 } - 400 } \\ { \text { Solve the equation. (Enter your answers } } \\ { x = } \end{array} \right.$$

Solve for x

$$x=20$$

$$x = \frac{3020}{9} = 335\frac{5}{9} \approx 335.555555556$$

Steps Using the Quadratic Formula

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

$$6400x-18x^{2}-400=120400$$

Subtract 120400 from both sides.

$$6400x-18x^{2}-400-120400=0$$

Subtract 120400 from -400 to get -120800.

$$6400x-18x^{2}-120800=0$$

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.

$$-18x^{2}+6400x-120800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -18 for a, 6400 for b, and -120800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6400±\sqrt{6400^{2}-4\left(-18\right)\left(-120800\right)}}{2\left(-18\right)}$$

Square 6400.

$$x=\frac{-6400±\sqrt{40960000-4\left(-18\right)\left(-120800\right)}}{2\left(-18\right)}$$

Multiply -4 times -18.

$$x=\frac{-6400±\sqrt{40960000+72\left(-120800\right)}}{2\left(-18\right)}$$

Multiply 72 times -120800.

$$x=\frac{-6400±\sqrt{40960000-8697600}}{2\left(-18\right)}$$

Add 40960000 to -8697600.

$$x=\frac{-6400±\sqrt{32262400}}{2\left(-18\right)}$$

Take the square root of 32262400.

$$x=\frac{-6400±5680}{2\left(-18\right)}$$

Multiply 2 times -18.

$$x=\frac{-6400±5680}{-36}$$

Now solve the equation x=\frac{-6400±5680}{-36} when ± is plus. Add -6400 to 5680.

$$x=-\frac{720}{-36}$$

Divide -720 by -36.

$$x=20$$

Now solve the equation x=\frac{-6400±5680}{-36} when ± is minus. Subtract 5680 from -6400.

$$x=-\frac{12080}{-36}$$

Reduce the fraction \frac{-12080}{-36} to lowest terms by extracting and canceling out 4.

$$x=\frac{3020}{9}$$

The equation is now solved.

$$x=20 x=\frac{3020}{9}$$

Steps for Completing the Square

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

$$6400x-18x^{2}-400=120400$$

Add 400 to both sides.

$$6400x-18x^{2}=120400+400$$

Add 120400 and 400 to get 120800.

$$6400x-18x^{2}=120800$$

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.

$$-18x^{2}+6400x=120800$$

Divide both sides by -18.

$$\frac{-18x^{2}+6400x}{-18}=\frac{120800}{-18}$$

Dividing by -18 undoes the multiplication by -18.

$$x^{2}+\frac{6400}{-18}x=\frac{120800}{-18}$$

Reduce the fraction \frac{6400}{-18} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{3200}{9}x=\frac{120800}{-18}$$

Reduce the fraction \frac{120800}{-18} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{3200}{9}x=-\frac{60400}{9}$$

Divide -\frac{3200}{9}, the coefficient of the x term, by 2 to get -\frac{1600}{9}. Then add the square of -\frac{1600}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3200}{9}x+\left(-\frac{1600}{9}\right)^{2}=-\frac{60400}{9}+\left(-\frac{1600}{9}\right)^{2}$$

Square -\frac{1600}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3200}{9}x+\frac{2560000}{81}=-\frac{60400}{9}+\frac{2560000}{81}$$

Add -\frac{60400}{9} to \frac{2560000}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3200}{9}x+\frac{2560000}{81}=\frac{2016400}{81}$$

Factor x^{2}-\frac{3200}{9}x+\frac{2560000}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1600}{9}\right)^{2}=\frac{2016400}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1600}{9}\right)^{2}}=\sqrt{\frac{2016400}{81}}$$

Simplify.

$$x-\frac{1600}{9}=\frac{1420}{9} x-\frac{1600}{9}=-\frac{1420}{9}$$

Add \frac{1600}{9} to both sides of the equation.

$$x=\frac{3020}{9} x=20$$