Factor the expression by grouping. First, the expression needs to be rewritten as $x^{2}+ax+bx+90$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-21$$ $$ab=1\times 90=90$$
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 $90$.
Rewrite $x^{2}-21x+90$ as $\left(x^{2}-15x\right)+\left(-6x+90\right)$.
$$\left(x^{2}-15x\right)+\left(-6x+90\right)$$
Factor out $x$ in the first and $-6$ in the second group.
$$x\left(x-15\right)-6\left(x-15\right)$$
Factor out common term $x-15$ by using distributive property.
$$\left(x-15\right)\left(x-6\right)$$
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 $ax^{2}+bx+c=0$.
$$x^{2}-21x+90=0$$
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.
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 $x^2+Bx+C=0$.
$$x ^ 2 -21x +90 = 0$$
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 $rs = C$
$$r + s = 21 $$ $$ rs = 90$$
Two numbers $r$ and $s$ sum up to $21$ exactly when the average of the two numbers is $\frac{1}{2}*21 = \frac{21}{2}$. 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 $u$.
$$r = \frac{21}{2} - u$$ $$s = \frac{21}{2} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = 90$
$$(\frac{21}{2} - u) (\frac{21}{2} + u) = 90$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{441}{4} - u^2 = 90$$
Simplify the expression by subtracting $\frac{441}{4}$ on both sides
$$-u^2 = 90-\frac{441}{4} = -\frac{81}{4}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
