Solve -4.9z^2+z+360=-4z+142

$$-4.9 { z }^{ 2 } +z+360=-4z+142$$

Solve for z

$$z = \frac{\sqrt{107445} + 25}{49} \approx 7.199755944$$

$$z=\frac{25-\sqrt{107445}}{49}\approx -6.179347781$$

Steps Using the Quadratic Formula

Add 4z to both sides.

$$-4.9z^{2}+z+360+4z=142$$

Combine z and 4z to get 5z.

$$-4.9z^{2}+5z+360=142$$

Subtract 142 from both sides.

$$-4.9z^{2}+5z+360-142=0$$

Subtract 142 from 360 to get 218.

$$-4.9z^{2}+5z+218=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 5 for b, and 218 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-5±\sqrt{5^{2}-4\left(-4.9\right)\times 218}}{2\left(-4.9\right)}$$

Square 5.

$$z=\frac{-5±\sqrt{25-4\left(-4.9\right)\times 218}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$z=\frac{-5±\sqrt{25+19.6\times 218}}{2\left(-4.9\right)}$$

Multiply 19.6 times 218.

$$z=\frac{-5±\sqrt{25+4272.8}}{2\left(-4.9\right)}$$

Add 25 to 4272.8.

$$z=\frac{-5±\sqrt{4297.8}}{2\left(-4.9\right)}$$

Take the square root of 4297.8.

$$z=\frac{-5±\frac{\sqrt{107445}}{5}}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$z=\frac{-5±\frac{\sqrt{107445}}{5}}{-9.8}$$

Now solve the equation z=\frac{-5±\frac{\sqrt{107445}}{5}}{-9.8} when ± is plus. Add -5 to \frac{\sqrt{107445}}{5}.

$$z=\frac{\frac{\sqrt{107445}}{5}-5}{-9.8}$$

Divide -5+\frac{\sqrt{107445}}{5} by -9.8 by multiplying -5+\frac{\sqrt{107445}}{5} by the reciprocal of -9.8.

$$z=\frac{25-\sqrt{107445}}{49}$$

Now solve the equation z=\frac{-5±\frac{\sqrt{107445}}{5}}{-9.8} when ± is minus. Subtract \frac{\sqrt{107445}}{5} from -5.

$$z=\frac{-\frac{\sqrt{107445}}{5}-5}{-9.8}$$

Divide -5-\frac{\sqrt{107445}}{5} by -9.8 by multiplying -5-\frac{\sqrt{107445}}{5} by the reciprocal of -9.8.

$$z=\frac{\sqrt{107445}+25}{49}$$

The equation is now solved.

$$z=\frac{25-\sqrt{107445}}{49} z=\frac{\sqrt{107445}+25}{49}$$

Steps for Completing the Square

Add 4z to both sides.

$$-4.9z^{2}+z+360+4z=142$$

Combine z and 4z to get 5z.

$$-4.9z^{2}+5z+360=142$$

Subtract 360 from both sides.

$$-4.9z^{2}+5z=142-360$$

Subtract 360 from 142 to get -218.

$$-4.9z^{2}+5z=-218$$

Divide both sides of the equation by -4.9, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-4.9z^{2}+5z}{-4.9}=-\frac{218}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$z^{2}+\frac{5}{-4.9}z=-\frac{218}{-4.9}$$

Divide 5 by -4.9 by multiplying 5 by the reciprocal of -4.9.

$$z^{2}-\frac{50}{49}z=-\frac{218}{-4.9}$$

Divide -218 by -4.9 by multiplying -218 by the reciprocal of -4.9.

$$z^{2}-\frac{50}{49}z=\frac{2180}{49}$$

Divide -\frac{50}{49}, the coefficient of the x term, by 2 to get -\frac{25}{49}. Then add the square of -\frac{25}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-\frac{50}{49}z+\left(-\frac{25}{49}\right)^{2}=\frac{2180}{49}+\left(-\frac{25}{49}\right)^{2}$$

Square -\frac{25}{49} by squaring both the numerator and the denominator of the fraction.

$$z^{2}-\frac{50}{49}z+\frac{625}{2401}=\frac{2180}{49}+\frac{625}{2401}$$

Add \frac{2180}{49} to \frac{625}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z^{2}-\frac{50}{49}z+\frac{625}{2401}=\frac{107445}{2401}$$

Factor z^{2}-\frac{50}{49}z+\frac{625}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-\frac{25}{49}\right)^{2}=\frac{107445}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-\frac{25}{49}\right)^{2}}=\sqrt{\frac{107445}{2401}}$$

Simplify.

$$z-\frac{25}{49}=\frac{\sqrt{107445}}{49} z-\frac{25}{49}=-\frac{\sqrt{107445}}{49}$$

Add \frac{25}{49} to both sides of the equation.

$$z=\frac{\sqrt{107445}+25}{49} z=\frac{25-\sqrt{107445}}{49}$$