$$\frac { 5 x ^ { 2 } + 3 x - 2 } { x ^ { 2 } - 1 } = 0$$
Solve for x
$x=\frac{2}{5}=0.4$
Steps Using Factoring By Grouping
Steps Using the Quadratic Formula
Steps for Completing the Square
Steps Using Factoring By Grouping
Variable $x$ cannot be equal to any of the values $-1,1$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-1\right)\left(x+1\right)$.
$$5x^{2}+3x-2=0$$
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as $5x^{2}+ax+bx-2$. To find $a$ and $b$, set up a system to be solved.
$$a+b=3$$ $$ab=5\left(-2\right)=-10$$
Since $ab$ is negative, $a$ and $b$ have the opposite signs. Since $a+b$ is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product $-10$.
$$-1,10$$ $$-2,5$$
Calculate the sum for each pair.
$$-1+10=9$$ $$-2+5=3$$
The solution is the pair that gives sum $3$.
$$a=-2$$ $$b=5$$
Rewrite $5x^{2}+3x-2$ as $\left(5x^{2}-2x\right)+\left(5x-2\right)$.
$$\left(5x^{2}-2x\right)+\left(5x-2\right)$$
Factor out $x$ in $5x^{2}-2x$.
$$x\left(5x-2\right)+5x-2$$
Factor out common term $5x-2$ by using distributive property.
$$\left(5x-2\right)\left(x+1\right)$$
To find equation solutions, solve $5x-2=0$ and $x+1=0$.
$$x=\frac{2}{5}$$ $$x=-1$$
Variable $x$ cannot be equal to $-1$.
$$x=\frac{2}{5}$$
Steps Using the Quadratic Formula
Variable $x$ cannot be equal to any of the values $-1,1$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-1\right)\left(x+1\right)$.
$$5x^{2}+3x-2=0$$
This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $5$ for $a$, $3$ for $b$, and $-2$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.
Now solve the equation $x=\frac{-3±7}{10}$ when $±$ is plus. Add $-3$ to $7$.
$$x=\frac{4}{10}$$
Reduce the fraction $\frac{4}{10}$ to lowest terms by extracting and canceling out $2$.
$$x=\frac{2}{5}$$
Now solve the equation $x=\frac{-3±7}{10}$ when $±$ is minus. Subtract $7$ from $-3$.
$$x=-\frac{10}{10}$$
Divide $-10$ by $10$.
$$x=-1$$
The equation is now solved.
$$x=\frac{2}{5}$$ $$x=-1$$
Variable $x$ cannot be equal to $-1$.
$$x=\frac{2}{5}$$
Steps for Completing the Square
Variable $x$ cannot be equal to any of the values $-1,1$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-1\right)\left(x+1\right)$.
$$5x^{2}+3x-2=0$$
Add $2$ to both sides. Anything plus zero gives itself.
$$5x^{2}+3x=2$$
Divide both sides by $5$.
$$\frac{5x^{2}+3x}{5}=\frac{2}{5}$$
Dividing by $5$ undoes the multiplication by $5$.
$$x^{2}+\frac{3}{5}x=\frac{2}{5}$$
Divide $\frac{3}{5}$, the coefficient of the $x$ term, by $2$ to get $\frac{3}{10}$. Then add the square of $\frac{3}{10}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.
Factor $x^{2}+\frac{3}{5}x+\frac{9}{100}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.
