Factor the expression by grouping. First, the expression needs to be rewritten as $a^{2}+pa+qa-2$. To find $p$ and $q$, set up a system to be solved.
$$p+q=1$$ $$pq=1\left(-2\right)=-2$$
Since $pq$ is negative, $p$ and $q$ have the opposite signs. Since $p+q$ is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.
$$p=-1$$ $$q=2$$
Rewrite $a^{2}+a-2$ as $\left(a^{2}-a\right)+\left(2a-2\right)$.
$$\left(a^{2}-a\right)+\left(2a-2\right)$$
Factor out $a$ in the first and $2$ in the second group.
$$a\left(a-1\right)+2\left(a-1\right)$$
Factor out common term $a-1$ by using distributive property.
$$\left(a-1\right)\left(a+2\right)$$
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 $ax^{2}+bx+c=0$.
$$a^{2}+a-2=0$$
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.
$$a=\frac{-1±\sqrt{1^{2}-4\left(-2\right)}}{2}$$
Square $1$.
$$a=\frac{-1±\sqrt{1-4\left(-2\right)}}{2}$$
Multiply $-4$ times $-2$.
$$a=\frac{-1±\sqrt{1+8}}{2}$$
Add $1$ to $8$.
$$a=\frac{-1±\sqrt{9}}{2}$$
Take the square root of $9$.
$$a=\frac{-1±3}{2}$$
Now solve the equation $a=\frac{-1±3}{2}$ when $±$ is plus. Add $-1$ to $3$.
$$a=\frac{2}{2}$$
Divide $2$ by $2$.
$$a=1$$
Now solve the equation $a=\frac{-1±3}{2}$ when $±$ is minus. Subtract $3$ from $-1$.
$$a=-\frac{4}{2}$$
Divide $-4$ by $2$.
$$a=-2$$
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 $-2$ for $x_{2}$.
Simplify all the expressions of the form $p-\left(-q\right)$ to $p+q$.
$$a^{2}+a-2=\left(a-1\right)\left(a+2\right)$$
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 $x^2+Bx+C=0$.
$$x ^ 2 +1x -2 = 0$$
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 $rs = C$
$$r + s = -1 $$ $$ rs = -2$$
Two numbers $r$ and $s$ sum up to $-1$ exactly when the average of the two numbers is $\frac{1}{2}*-1 = -\frac{1}{2}$. 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 $u$.
$$r = -\frac{1}{2} - u$$ $$s = -\frac{1}{2} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -2$
$$(-\frac{1}{2} - u) (-\frac{1}{2} + u) = -2$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{1}{4} - u^2 = -2$$
Simplify the expression by subtracting $\frac{1}{4}$ on both sides
$$-u^2 = -2-\frac{1}{4} = -\frac{9}{4}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
