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 $ax^{2}+bx+c=0$.
$$b^{2}+7b-24=0$$
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.
$$b=\frac{-7±\sqrt{7^{2}-4\left(-24\right)}}{2}$$
Square $7$.
$$b=\frac{-7±\sqrt{49-4\left(-24\right)}}{2}$$
Multiply $-4$ times $-24$.
$$b=\frac{-7±\sqrt{49+96}}{2}$$
Add $49$ to $96$.
$$b=\frac{-7±\sqrt{145}}{2}$$
Now solve the equation $b=\frac{-7±\sqrt{145}}{2}$ when $±$ is plus. Add $-7$ to $\sqrt{145}$.
$$b=\frac{\sqrt{145}-7}{2}$$
Now solve the equation $b=\frac{-7±\sqrt{145}}{2}$ when $±$ is minus. Subtract $\sqrt{145}$ from $-7$.
$$b=\frac{-\sqrt{145}-7}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{-7+\sqrt{145}}{2}$ for $x_{1}$ and $\frac{-7-\sqrt{145}}{2}$ for $x_{2}$.
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 $x^2+Bx+C=0$.
$$x ^ 2 +7x -24 = 0$$
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 $rs = C$
$$r + s = -7 $$ $$ rs = -24$$
Two numbers $r$ and $s$ sum up to $-7$ exactly when the average of the two numbers is $\frac{1}{2}*-7 = -\frac{7}{2}$. 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 $u$.
$$r = -\frac{7}{2} - u$$ $$s = -\frac{7}{2} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -24$
$$(-\frac{7}{2} - u) (-\frac{7}{2} + u) = -24$$
Simplify by expanding $(a -b) (a + b) = a^2 – b^2$
$$\frac{49}{4} - u^2 = -24$$
Simplify the expression by subtracting $\frac{49}{4}$ on both sides
$$-u^2 = -24-\frac{49}{4} = -\frac{145}{4}$$
Simplify the expression by multiplying $-1$ on both sides and take the square root to obtain the value of unknown variable $u$
