Factor the expression by grouping. First, the expression needs to be rewritten as $x^{2}+ax+bx+42$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-13$$ $$ab=1\times 42=42$$
Since $ab$ is positive, $a$ and $b$ have the same sign. Since $a+b$ is negative, $a$ and $b$ are both negative. List all such integer pairs that give product $42$.
Rewrite $x^{2}-13x+42$ as $\left(x^{2}-7x\right)+\left(-6x+42\right)$.
$$\left(x^{2}-7x\right)+\left(-6x+42\right)$$
Factor out $x$ in the first and $-6$ in the second group.
$$x\left(x-7\right)-6\left(x-7\right)$$
Factor out common term $x-7$ by using distributive property.
$$\left(x-7\right)\left(x-6\right)$$
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 $ax^{2}+bx+c=0$.
$$x^{2}-13x+42=0$$
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.
Now solve the equation $x=\frac{13±1}{2}$ when $±$ is plus. Add $13$ to $1$.
$$x=\frac{14}{2}$$
Divide $14$ by $2$.
$$x=7$$
Now solve the equation $x=\frac{13±1}{2}$ when $±$ is minus. Subtract $1$ from $13$.
$$x=\frac{12}{2}$$
Divide $12$ by $2$.
$$x=6$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $7$ for $x_{1}$ and $6$ for $x_{2}$.
$$x^{2}-13x+42=\left(x-7\right)\left(x-6\right)$$
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 $x^2+Bx+C=0$.
$$x ^ 2 -13x +42 = 0$$
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 $rs = C$
$$r + s = 13 $$ $$ rs = 42$$
Two numbers $r$ and $s$ sum up to $13$ exactly when the average of the two numbers is $\frac{1}{2}*13 = \frac{13}{2}$. 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 $u$.
$$r = \frac{13}{2} - u$$ $$s = \frac{13}{2} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = 42$
$$(\frac{13}{2} - u) (\frac{13}{2} + u) = 42$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{169}{4} - u^2 = 42$$
Simplify the expression by subtracting $\frac{169}{4}$ on both sides
$$-u^2 = 42-\frac{169}{4} = -\frac{1}{4}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
