Solve 5z^2=9-44z

$$5 z ^ { 2 } = 9 - 44 z$$

Solve for z

$$z=-9$$

$$z=\frac{1}{5}=0.2$$

Steps Using Factoring By Grouping

Subtract 9 from both sides.

$$5z^{2}-9=-44z$$

Add 44z to both sides.

$$5z^{2}-9+44z=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$5z^{2}+44z-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5z^{2}+az+bz-9. To find a and b, set up a system to be solved.

$$a+b=44 ab=5\left(-9\right)=-45$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -45.

$$-1,45 -3,15 -5,9$$

Calculate the sum for each pair.

$$-1+45=44 -3+15=12 -5+9=4$$

The solution is the pair that gives sum 44.

$$a=-1 b=45$$

Rewrite 5z^{2}+44z-9 as \left(5z^{2}-z\right)+\left(45z-9\right).

$$\left(5z^{2}-z\right)+\left(45z-9\right)$$

Factor out z in the first and 9 in the second group.

$$z\left(5z-1\right)+9\left(5z-1\right)$$

Factor out common term 5z-1 by using distributive property.

$$\left(5z-1\right)\left(z+9\right)$$

To find equation solutions, solve 5z-1=0 and z+9=0.

$$z=\frac{1}{5} z=-9$$

Steps Using the Quadratic Formula

Subtract 9 from both sides.

$$5z^{2}-9=-44z$$

Add 44z to both sides.

$$5z^{2}-9+44z=0$$

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.

$$5z^{2}+44z-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 44 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-44±\sqrt{44^{2}-4\times 5\left(-9\right)}}{2\times 5}$$

Square 44.

$$z=\frac{-44±\sqrt{1936-4\times 5\left(-9\right)}}{2\times 5}$$

Multiply -4 times 5.

$$z=\frac{-44±\sqrt{1936-20\left(-9\right)}}{2\times 5}$$

Multiply -20 times -9.

$$z=\frac{-44±\sqrt{1936+180}}{2\times 5}$$

Add 1936 to 180.

$$z=\frac{-44±\sqrt{2116}}{2\times 5}$$

Take the square root of 2116.

$$z=\frac{-44±46}{2\times 5}$$

Multiply 2 times 5.

$$z=\frac{-44±46}{10}$$

Now solve the equation z=\frac{-44±46}{10} when ± is plus. Add -44 to 46.

$$z=\frac{2}{10}$$

Reduce the fraction \frac{2}{10} to lowest terms by extracting and canceling out 2.

$$z=\frac{1}{5}$$

Now solve the equation z=\frac{-44±46}{10} when ± is minus. Subtract 46 from -44.

$$z=-\frac{90}{10}$$

Divide -90 by 10.

$$z=-9$$

The equation is now solved.

$$z=\frac{1}{5} z=-9$$

Steps for Completing the Square

Add 44z to both sides.

$$5z^{2}+44z=9$$

Divide both sides by 5.

$$\frac{5z^{2}+44z}{5}=\frac{9}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$z^{2}+\frac{44}{5}z=\frac{9}{5}$$

Divide \frac{44}{5}, the coefficient of the x term, by 2 to get \frac{22}{5}. Then add the square of \frac{22}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}+\frac{44}{5}z+\left(\frac{22}{5}\right)^{2}=\frac{9}{5}+\left(\frac{22}{5}\right)^{2}$$

Square \frac{22}{5} by squaring both the numerator and the denominator of the fraction.

$$z^{2}+\frac{44}{5}z+\frac{484}{25}=\frac{9}{5}+\frac{484}{25}$$

Add \frac{9}{5} to \frac{484}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$z^{2}+\frac{44}{5}z+\frac{484}{25}=\frac{529}{25}$$

Factor z^{2}+\frac{44}{5}z+\frac{484}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z+\frac{22}{5}\right)^{2}=\frac{529}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z+\frac{22}{5}\right)^{2}}=\sqrt{\frac{529}{25}}$$

Simplify.

$$z+\frac{22}{5}=\frac{23}{5} z+\frac{22}{5}=-\frac{23}{5}$$

Subtract \frac{22}{5} from both sides of the equation.

$$z=\frac{1}{5} z=-9$$