Factor the expression by grouping. First, the expression needs to be rewritten as $10x^{2}+ax+bx-7$. To find $a$ and $b$, set up a system to be solved.
$$a+b=-9$$ $$ab=10\left(-7\right)=-70$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product $-70$.
$$1,-70$$ $$2,-35$$ $$5,-14$$ $$7,-10$$
Calculate the sum for each pair.
$$1-70=-69$$ $$2-35=-33$$ $$5-14=-9$$ $$7-10=-3$$
The solution is the pair that gives sum $-9$.
$$a=-14$$ $$b=5$$
Rewrite $10x^{2}-9x-7$ as $\left(10x^{2}-14x\right)+\left(5x-7\right)$.
$$\left(10x^{2}-14x\right)+\left(5x-7\right)$$
Factor out $2x$ in $10x^{2}-14x$.
$$2x\left(5x-7\right)+5x-7$$
Factor out common term $5x-7$ by using distributive property.
$$\left(5x-7\right)\left(2x+1\right)$$
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 $ax^{2}+bx+c=0$.
$$10x^{2}-9x-7=0$$
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 $x=\frac{9±19}{20}$ when $±$ is plus. Add $9$ to $19$.
$$x=\frac{28}{20}$$
Reduce the fraction $\frac{28}{20}$ to lowest terms by extracting and canceling out $4$.
$$x=\frac{7}{5}$$
Now solve the equation $x=\frac{9±19}{20}$ when $±$ is minus. Subtract $19$ from $9$.
$$x=-\frac{10}{20}$$
Reduce the fraction $\frac{-10}{20}$ to lowest terms by extracting and canceling out $10$.
$$x=-\frac{1}{2}$$
Factor the original expression using $ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)$. Substitute $\frac{7}{5}$ for $x_{1}$ and $-\frac{1}{2}$ for $x_{2}$.
Multiply $\frac{5x-7}{5}$ times $\frac{2x+1}{2}$ 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 $10$
$$x ^ 2 -\frac{9}{10}x -\frac{7}{10} = 0$$
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 $rs = C$
$$r + s = \frac{9}{10} $$ $$ rs = -\frac{7}{10}$$
Two numbers $r$ and $s$ sum up to $\frac{9}{10}$ exactly when the average of the two numbers is $\frac{1}{2}*\frac{9}{10} = \frac{9}{20}$. 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 $u$.
$$r = \frac{9}{20} - u$$ $$s = \frac{9}{20} + u$$
To solve for unknown quantity $u$, substitute these in the product equation $rs = -\frac{7}{10}$
