Solve 1/4x^2+80x-1700=0

$$\frac{ 1 }{ 4 } { x }^{ 2 } +80x-1700 = 0$$

Solve for x

$$x=-340$$

$$x=20$$

Steps Using the Quadratic Formula

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.

$$\frac{1}{4}x^{2}+80x-1700=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{4} for a, 80 for b, and -1700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-80±\sqrt{80^{2}-4\times \frac{1}{4}\left(-1700\right)}}{2\times \frac{1}{4}}$$

Square 80.

$$x=\frac{-80±\sqrt{6400-4\times \frac{1}{4}\left(-1700\right)}}{2\times \frac{1}{4}}$$

Multiply -4 times \frac{1}{4}.

$$x=\frac{-80±\sqrt{6400-\left(-1700\right)}}{2\times \frac{1}{4}}$$

Multiply -1 times -1700.

$$x=\frac{-80±\sqrt{6400+1700}}{2\times \frac{1}{4}}$$

Add 6400 to 1700.

$$x=\frac{-80±\sqrt{8100}}{2\times \frac{1}{4}}$$

Take the square root of 8100.

$$x=\frac{-80±90}{2\times \frac{1}{4}}$$

Multiply 2 times \frac{1}{4}.

$$x=\frac{-80±90}{\frac{1}{2}}$$

Now solve the equation x=\frac{-80±90}{\frac{1}{2}} when ± is plus. Add -80 to 90.

$$x=\frac{10}{\frac{1}{2}}$$

Divide 10 by \frac{1}{2} by multiplying 10 by the reciprocal of \frac{1}{2}.

$$x=20$$

Now solve the equation x=\frac{-80±90}{\frac{1}{2}} when ± is minus. Subtract 90 from -80.

$$x=-\frac{170}{\frac{1}{2}}$$

Divide -170 by \frac{1}{2} by multiplying -170 by the reciprocal of \frac{1}{2}.

$$x=-340$$

The equation is now solved.

$$x=20 x=-340$$

Steps for Completing the Square

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.

$$\frac{1}{4}x^{2}+80x-1700=0$$

Add 1700 to both sides of the equation.

$$\frac{1}{4}x^{2}+80x-1700-\left(-1700\right)=-\left(-1700\right)$$

Subtracting -1700 from itself leaves 0.

$$\frac{1}{4}x^{2}+80x=-\left(-1700\right)$$

Subtract -1700 from 0.

$$\frac{1}{4}x^{2}+80x=1700$$

Multiply both sides by 4.

$$\frac{\frac{1}{4}x^{2}+80x}{\frac{1}{4}}=\frac{1700}{\frac{1}{4}}$$

Dividing by \frac{1}{4} undoes the multiplication by \frac{1}{4}.

$$x^{2}+\frac{80}{\frac{1}{4}}x=\frac{1700}{\frac{1}{4}}$$

Divide 80 by \frac{1}{4} by multiplying 80 by the reciprocal of \frac{1}{4}.

$$x^{2}+320x=\frac{1700}{\frac{1}{4}}$$

Divide 1700 by \frac{1}{4} by multiplying 1700 by the reciprocal of \frac{1}{4}.

$$x^{2}+320x=6800$$

Divide 320, the coefficient of the x term, by 2 to get 160. Then add the square of 160 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+320x+160^{2}=6800+160^{2}$$

Square 160.

$$x^{2}+320x+25600=6800+25600$$

Add 6800 to 25600.

$$x^{2}+320x+25600=32400$$

Factor x^{2}+320x+25600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+160\right)^{2}=32400$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+160\right)^{2}}=\sqrt{32400}$$

Simplify.

$$x+160=180 x+160=-180$$

Subtract 160 from both sides of the equation.

$$x=20 x=-340$$