Solve y^2+8y+16=0

$$y ^ { 2 } + 8 y + 16 = 0$$

Solve for y

$$y=-4$$

Steps Using Factoring

To solve the equation, factor y^{2}+8y+16 using formula y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). To find a and b, set up a system to be solved.

$$a+b=8 ab=16$$

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 16.

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

Calculate the sum for each pair.

$$1+16=17 2+8=10 4+4=8$$

The solution is the pair that gives sum 8.

$$a=4 b=4$$

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

$$\left(y+4\right)\left(y+4\right)$$

Rewrite as a binomial square.

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

To find equation solution, solve y+4=0.

$$y=-4$$

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 y^{2}+ay+by+16. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 16=16$$

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 16.

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

Calculate the sum for each pair.

$$1+16=17 2+8=10 4+4=8$$

The solution is the pair that gives sum 8.

$$a=4 b=4$$

Rewrite y^{2}+8y+16 as \left(y^{2}+4y\right)+\left(4y+16\right).

$$\left(y^{2}+4y\right)+\left(4y+16\right)$$

Factor out y in the first and 4 in the second group.

$$y\left(y+4\right)+4\left(y+4\right)$$

Factor out common term y+4 by using distributive property.

$$\left(y+4\right)\left(y+4\right)$$

Rewrite as a binomial square.

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

To find equation solution, solve y+4=0.

$$y=-4$$

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.

$$y^{2}+8y+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-8±\sqrt{8^{2}-4\times 16}}{2}$$

Square 8.

$$y=\frac{-8±\sqrt{64-4\times 16}}{2}$$

Multiply -4 times 16.

$$y=\frac{-8±\sqrt{64-64}}{2}$$

Add 64 to -64.

$$y=\frac{-8±\sqrt{0}}{2}$$

Take the square root of 0.

$$y=-\frac{8}{2}$$

Divide -8 by 2.

$$y=-4$$

Steps for Completing the Square

Factor y^{2}+8y+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+4\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+4\right)^{2}}=\sqrt{0}$$

Simplify.

$$y+4=0 y+4=0$$

Subtract 4 from both sides of the equation.

$$y=-4 y=-4$$

The equation is now solved. Solutions are the same.

$$y=-4$$

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.

$$x ^ 2 +8x +16 = 0$$

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

$$r + s = -8 rs = 16$$

Two numbers r and s sum up to -8 exactly when the average of the two numbers is \frac{1}{2}*-8 = -4. 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>

$$r = -4 - u s = -4 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 16

$$(-4 - u) (-4 + u) = 16$$

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

$$16 - u^2 = 16$$

Simplify the expression by subtracting 16 on both sides

$$-u^2 = 16-16 = 0$$

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.

$$r = s = -4$$