Factor the expression by grouping. First, the expression needs to be rewritten as $10z^{2}+az+bz+10$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-29$$ $$ab=10\times 10=100$$
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 $100$.
Rewrite $10z^{2}-29z+10$ as $\left(10z^{2}-25z\right)+\left(-4z+10\right)$.
$$\left(10z^{2}-25z\right)+\left(-4z+10\right)$$
Factor out $5z$ in the first and $-2$ in the second group.
$$5z\left(2z-5\right)-2\left(2z-5\right)$$
Factor out common term $2z-5$ by using distributive property.
$$\left(2z-5\right)\left(5z-2\right)$$
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 $ax^{2}+bx+c=0$.
$$10z^{2}-29z+10=0$$
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 $z=\frac{29±21}{20}$ when $±$ is plus. Add $29$ to $21$.
$$z=\frac{50}{20}$$
Reduce the fraction $\frac{50}{20}$ to lowest terms by extracting and canceling out $10$.
$$z=\frac{5}{2}$$
Now solve the equation $z=\frac{29±21}{20}$ when $±$ is minus. Subtract $21$ from $29$.
$$z=\frac{8}{20}$$
Reduce the fraction $\frac{8}{20}$ to lowest terms by extracting and canceling out $4$.
$$z=\frac{2}{5}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{5}{2}$ for $x_{1}$ and $\frac{2}{5}$ for $x_{2}$.
Multiply $\frac{2z-5}{2}$ times $\frac{5z-2}{5}$ by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.
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 $x^2+Bx+C=0$.This is achieved by dividing both sides of the equation by $10$
$$x ^ 2 -\frac{29}{10}x +1 = 0$$
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 $rs = C$
$$r + s = \frac{29}{10} $$ $$ rs = 1$$
Two numbers $r$ and $s$ sum up to $\frac{29}{10}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{29}{10} = \frac{29}{20}$. 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 $u$.
