Solve z^2-16=6z

$$z ^ { 2 } - 16 = 6 z$$

Solve for z

$$z=-2$$

$$z=8$$

Steps Using Factoring

Subtract 6z from both sides.

$$z^{2}-16-6z=0$$

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

$$z^{2}-6z-16=0$$

To solve the equation, factor z^{2}-6z-16 using formula z^{2}+\left(a+b\right)z+ab=\left(z+a\right)\left(z+b\right). To find a and b, set up a system to be solved.

$$a+b=-6 ab=-16$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -6.

$$a=-8 b=2$$

Rewrite factored expression \left(z+a\right)\left(z+b\right) using the obtained values.

$$\left(z-8\right)\left(z+2\right)$$

To find equation solutions, solve z-8=0 and z+2=0.

$$z=8 z=-2$$

Steps Using Factoring By Grouping

Subtract 6z from both sides.

$$z^{2}-16-6z=0$$

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

$$z^{2}-6z-16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as z^{2}+az+bz-16. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-16\right)=-16$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -6.

$$a=-8 b=2$$

Rewrite z^{2}-6z-16 as \left(z^{2}-8z\right)+\left(2z-16\right).

$$\left(z^{2}-8z\right)+\left(2z-16\right)$$

Factor out z in the first and 2 in the second group.

$$z\left(z-8\right)+2\left(z-8\right)$$

Factor out common term z-8 by using distributive property.

$$\left(z-8\right)\left(z+2\right)$$

To find equation solutions, solve z-8=0 and z+2=0.

$$z=8 z=-2$$

Steps Using the Quadratic Formula

Subtract 6z from both sides.

$$z^{2}-16-6z=0$$

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.

$$z^{2}-6z-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-16\right)}}{2}$$

Square -6.

$$z=\frac{-\left(-6\right)±\sqrt{36-4\left(-16\right)}}{2}$$

Multiply -4 times -16.

$$z=\frac{-\left(-6\right)±\sqrt{36+64}}{2}$$

Add 36 to 64.

$$z=\frac{-\left(-6\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$z=\frac{-\left(-6\right)±10}{2}$$

The opposite of -6 is 6.

$$z=\frac{6±10}{2}$$

Now solve the equation z=\frac{6±10}{2} when ± is plus. Add 6 to 10.

$$z=\frac{16}{2}$$

Divide 16 by 2.

$$z=8$$

Now solve the equation z=\frac{6±10}{2} when ± is minus. Subtract 10 from 6.

$$z=-\frac{4}{2}$$

Divide -4 by 2.

$$z=-2$$

The equation is now solved.

$$z=8 z=-2$$

Steps for Completing the Square

Subtract 6z from both sides.

$$z^{2}-16-6z=0$$

Add 16 to both sides. Anything plus zero gives itself.

$$z^{2}-6z=16$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$z^{2}-6z+\left(-3\right)^{2}=16+\left(-3\right)^{2}$$

Square -3.

$$z^{2}-6z+9=16+9$$

Add 16 to 9.

$$z^{2}-6z+9=25$$

Factor z^{2}-6z+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-3\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-3\right)^{2}}=\sqrt{25}$$

Simplify.

$$z-3=5 z-3=-5$$

Add 3 to both sides of the equation.

$$z=8 z=-2$$