Solve z^2+6z+9=0

$$z ^ { 2 } + 6 z + 9 = 0$$

Solve for z

$$z=-3$$

Steps Using Factoring

To solve the equation, factor z^{2}+6z+9 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.

$$a+b=6 ab=9$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 9.

$$1,9 3,3$$

Calculate the sum for each pair.

$$1+9=10 3+3=6$$

The solution is the pair that gives sum 6.

$$a=3 b=3$$

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

$$\left(z+3\right)\left(z+3\right)$$

Rewrite as a binomial square.

$$\left(z+3\right)^{2}$$

To find equation solution, solve z+3=0.

$$z=-3$$

Steps Using Factoring By Grouping

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

$$a+b=6 ab=1\times 9=9$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 9.

$$1,9 3,3$$

Calculate the sum for each pair.

$$1+9=10 3+3=6$$

The solution is the pair that gives sum 6.

$$a=3 b=3$$

Rewrite z^{2}+6z+9 as \left(z^{2}+3z\right)+\left(3z+9\right).

$$\left(z^{2}+3z\right)+\left(3z+9\right)$$

Factor out z in the first and 3 in the second group.

$$z\left(z+3\right)+3\left(z+3\right)$$

Factor out common term z+3 by using distributive property.

$$\left(z+3\right)\left(z+3\right)$$

Rewrite as a binomial square.

$$\left(z+3\right)^{2}$$

To find equation solution, solve z+3=0.

$$z=-3$$

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.

$$z^{2}+6z+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-6±\sqrt{6^{2}-4\times 9}}{2}$$

Square 6.

$$z=\frac{-6±\sqrt{36-4\times 9}}{2}$$

Multiply -4 times 9.

$$z=\frac{-6±\sqrt{36-36}}{2}$$

Add 36 to -36.

$$z=\frac{-6±\sqrt{0}}{2}$$

Take the square root of 0.

$$z=-\frac{6}{2}$$

Divide -6 by 2.

$$z=-3$$

Steps for Completing the Square

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}.

$$\left(z+3\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z+3\right)^{2}}=\sqrt{0}$$

Simplify.

$$z+3=0 z+3=0$$

Subtract 3 from both sides of the equation.

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

The equation is now solved. Solutions are the same.

$$z=-3$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +6x +9 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -6 rs = 9$$

Two numbers r and s sum up to -6 exactly when the average of the two numbers is \frac{1}{2}*-6 = -3. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -3 - u s = -3 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 9

$$(-3 - u) (-3 + u) = 9$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$9 - u^2 = 9$$

Simplify the expression by subtracting 9 on both sides

$$-u^2 = 9-9 = 0$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 0 u = 0$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r = s = -3$$