Solve v^2+4v+4=0

$$v ^ { 2 } + 4 v + 4 = 0$$

Solve for v

$$v=-2$$

Steps Using Factoring

To solve the equation, factor v^{2}+4v+4 using formula v^{2}+\left(a+b\right)v+ab=\left(v+a\right)\left(v+b\right). To find a and b, set up a system to be solved.

$$a+b=4 ab=4$$

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 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 4.

$$a=2 b=2$$

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

$$\left(v+2\right)\left(v+2\right)$$

Rewrite as a binomial square.

$$\left(v+2\right)^{2}$$

To find equation solution, solve v+2=0.

$$v=-2$$

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 v^{2}+av+bv+4. To find a and b, set up a system to be solved.

$$a+b=4 ab=1\times 4=4$$

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 4.

$$1,4 2,2$$

Calculate the sum for each pair.

$$1+4=5 2+2=4$$

The solution is the pair that gives sum 4.

$$a=2 b=2$$

Rewrite v^{2}+4v+4 as \left(v^{2}+2v\right)+\left(2v+4\right).

$$\left(v^{2}+2v\right)+\left(2v+4\right)$$

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

$$v\left(v+2\right)+2\left(v+2\right)$$

Factor out common term v+2 by using distributive property.

$$\left(v+2\right)\left(v+2\right)$$

Rewrite as a binomial square.

$$\left(v+2\right)^{2}$$

To find equation solution, solve v+2=0.

$$v=-2$$

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.

$$v^{2}+4v+4=0$$

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

$$v=\frac{-4±\sqrt{4^{2}-4\times 4}}{2}$$

Square 4.

$$v=\frac{-4±\sqrt{16-4\times 4}}{2}$$

Multiply -4 times 4.

$$v=\frac{-4±\sqrt{16-16}}{2}$$

Add 16 to -16.

$$v=\frac{-4±\sqrt{0}}{2}$$

Take the square root of 0.

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

Divide -4 by 2.

$$v=-2$$

Steps for Completing the Square

Factor v^{2}+4v+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$v+2=0 v+2=0$$

Subtract 2 from both sides of the equation.

$$v=-2 v=-2$$

The equation is now solved. Solutions are the same.

$$v=-2$$

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.

$$x ^ 2 +4x +4 = 0$$

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

$$r + s = -4 rs = 4$$

Two numbers r and s sum up to -4 exactly when the average of the two numbers is \frac{1}{2}*-4 = -2. 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>

$$r = -2 - u s = -2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 4

$$(-2 - u) (-2 + u) = 4$$

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

$$4 - u^2 = 4$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = 4-4 = 0$$

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.

$$r = s = -2$$