Factor the expression by grouping. First, the expression needs to be rewritten as $x^{2}+ax+bx+7$. To find $a$ and $b$, set up a system to be solved.
$$a+b=8$$ $$ab=1\times 7=7$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is positive, $a$ and $b$ are both positive. The only such pair is the system solution.
$$a=1$$ $$b=7$$
Rewrite $x^{2}+8x+7$ as $\left(x^{2}+x\right)+\left(7x+7\right)$.
$$\left(x^{2}+x\right)+\left(7x+7\right)$$
Factor out $x$ in the first and $7$ in the second group.
$$x\left(x+1\right)+7\left(x+1\right)$$
Factor out common term $x+1$ by using distributive property.
$$\left(x+1\right)\left(x+7\right)$$
Steps Using the Quadratic Formula
Quadratic polynomial can be factored using the transformation $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$, where $x_{1}$ and $x_{2}$ are the solutions of the quadratic equation $ax^{2}+bx+c=0$.
$$x^{2}+8x+7=0$$
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.
$$x=\frac{-8±\sqrt{8^{2}-4\times 7}}{2}$$
Square $8$.
$$x=\frac{-8±\sqrt{64-4\times 7}}{2}$$
Multiply $-4$ times $7$.
$$x=\frac{-8±\sqrt{64-28}}{2}$$
Add $64$ to $-28$.
$$x=\frac{-8±\sqrt{36}}{2}$$
Take the square root of $36$.
$$x=\frac{-8±6}{2}$$
Now solve the equation $x=\frac{-8±6}{2}$ when $±$ is plus. Add $-8$ to $6$.
$$x=-\frac{2}{2}$$
Divide $-2$ by $2$.
$$x=-1$$
Now solve the equation $x=\frac{-8±6}{2}$ when $±$ is minus. Subtract $6$ from $-8$.
$$x=-\frac{14}{2}$$
Divide $-14$ by $2$.
$$x=-7$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $-1$ for $x_{1}$ and $-7$ for $x_{2}$.
Simplify all the expressions of the form $p-\left(-q\right)$ to $p+q$.
$$x^{2}+8x+7=\left(x+1\right)\left(x+7\right)$$
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 $x^2+Bx+C=0$.
$$x ^ 2 +8x +7 = 0$$
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 $rs = C$
$$r + s = -8 $$ $$ rs = 7$$
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 $rs = 7$
$$(-4 - u) (-4 + u) = 7$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$16 - u^2 = 7$$
Simplify the expression by subtracting $16$ on both sides
$$-u^2 = 7-16 = -9$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
$$u^2 = 9$$ $$u = \pm\sqrt{9} = \pm 3 $$
The factors $r$ and $s$ are the solutions to the quadratic equation. Substitute the value of $u$ to compute the $r$ and $s$.
