Solve 4z^2+160z=600

$$4 { z }^{ 2 } +160z = 600$$

Solve for z

$$z=5\sqrt{22}-20\approx 3.452078799$$

$$z=-5\sqrt{22}-20\approx -43.452078799$$

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.

$$4z^{2}+160z=600$$

Subtract 600 from both sides of the equation.

$$4z^{2}+160z-600=600-600$$

Subtracting 600 from itself leaves 0.

$$4z^{2}+160z-600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 160 for b, and -600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-160±\sqrt{160^{2}-4\times 4\left(-600\right)}}{2\times 4}$$

Square 160.

$$z=\frac{-160±\sqrt{25600-4\times 4\left(-600\right)}}{2\times 4}$$

Multiply -4 times 4.

$$z=\frac{-160±\sqrt{25600-16\left(-600\right)}}{2\times 4}$$

Multiply -16 times -600.

$$z=\frac{-160±\sqrt{25600+9600}}{2\times 4}$$

Add 25600 to 9600.

$$z=\frac{-160±\sqrt{35200}}{2\times 4}$$

Take the square root of 35200.

$$z=\frac{-160±40\sqrt{22}}{2\times 4}$$

Multiply 2 times 4.

$$z=\frac{-160±40\sqrt{22}}{8}$$

Now solve the equation z=\frac{-160±40\sqrt{22}}{8} when ± is plus. Add -160 to 40\sqrt{22}.

$$z=\frac{40\sqrt{22}-160}{8}$$

Divide -160+40\sqrt{22} by 8.

$$z=5\sqrt{22}-20$$

Now solve the equation z=\frac{-160±40\sqrt{22}}{8} when ± is minus. Subtract 40\sqrt{22} from -160.

$$z=\frac{-40\sqrt{22}-160}{8}$$

Divide -160-40\sqrt{22} by 8.

$$z=-5\sqrt{22}-20$$

The equation is now solved.

$$z=5\sqrt{22}-20 z=-5\sqrt{22}-20$$

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.

$$4z^{2}+160z=600$$

Divide both sides by 4.

$$\frac{4z^{2}+160z}{4}=\frac{600}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$z^{2}+\frac{160}{4}z=\frac{600}{4}$$

Divide 160 by 4.

$$z^{2}+40z=\frac{600}{4}$$

Divide 600 by 4.

$$z^{2}+40z=150$$

Divide 40, the coefficient of the x term, by 2 to get 20. Then add the square of 20 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}+40z+20^{2}=150+20^{2}$$

Square 20.

$$z^{2}+40z+400=150+400$$

Add 150 to 400.

$$z^{2}+40z+400=550$$

Factor z^{2}+40z+400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z+20\right)^{2}=550$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z+20\right)^{2}}=\sqrt{550}$$

Simplify.

$$z+20=5\sqrt{22} z+20=-5\sqrt{22}$$

Subtract 20 from both sides of the equation.

$$z=5\sqrt{22}-20 z=-5\sqrt{22}-20$$