To solve the equation, factor $t^{2}+8t+16$ using formula $t^{2}+\left(a+b\right)t+ab=\left(t+a\right)\left(t+b\right)$. 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 $\left(t+a\right)\left(t+b\right)$ using the obtained values.
$$\left(t+4\right)\left(t+4\right)$$
Rewrite as a binomial square.
$$\left(t+4\right)^{2}$$
To find equation solution, solve $t+4=0$.
$$t=-4$$
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 $t^{2}+at+bt+16$. 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 $t^{2}+8t+16$ as $\left(t^{2}+4t\right)+\left(4t+16\right)$.
$$\left(t^{2}+4t\right)+\left(4t+16\right)$$
Factor out $t$ in the first and $4$ in the second group.
$$t\left(t+4\right)+4\left(t+4\right)$$
Factor out common term $t+4$ by using distributive property.
$$\left(t+4\right)\left(t+4\right)$$
Rewrite as a binomial square.
$$\left(t+4\right)^{2}$$
To find equation solution, solve $t+4=0$.
$$t=-4$$
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.
$$t^{2}+8t+16=0$$
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, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.
$$t=\frac{-8±\sqrt{8^{2}-4\times 16}}{2}$$
Square $8$.
$$t=\frac{-8±\sqrt{64-4\times 16}}{2}$$
Multiply $-4$ times $16$.
$$t=\frac{-8±\sqrt{64-64}}{2}$$
Add $64$ to $-64$.
$$t=\frac{-8±\sqrt{0}}{2}$$
Take the square root of $0$.
$$t=-\frac{8}{2}$$
Divide $-8$ by $2$.
$$t=-4$$
Steps for Completing the Square
Factor $t^{2}+8t+16$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.
$$\left(t+4\right)^{2}=0$$
Take the square root of both sides of the equation.
$$\sqrt{\left(t+4\right)^{2}}=\sqrt{0}$$
Simplify.
$$t+4=0$$ $$t+4=0$$
Subtract $4$ from both sides of the equation.
$$t=-4$$ $$t=-4$$
The equation is now solved. Solutions are the same.
$$t=-4$$
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 $u$.
$$r = -4 - u$$ $$s = -4 + u$$
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 $s$.
