$$\frac { x ( 2 x - ( 7 - 5 x ) ) } { 7 x ( 9 x - ( 3 + 4 x ) ) } = \frac { 1 } { 6 }$$
Solve for x
$x=3$
Steps Using Factoring
Steps Using the Quadratic Formula
Steps for Completing the Square
Steps Using Factoring
Variable $x$ cannot be equal to any of the values $0,\frac{3}{5}$ since division by zero is not defined. Multiply both sides of the equation by $42x\left(5x-3\right)$, the least common multiple of $7x\left(9x-\left(3+4x\right)\right),6$.
To find the opposite of $7-5x$, find the opposite of each term.
$$6x\left(2x-7+5x\right)=7x\left(5x-3\right)$$
Combine $2x$ and $5x$ to get $7x$.
$$6x\left(7x-7\right)=7x\left(5x-3\right)$$
Use the distributive property to multiply $6x$ by $7x-7$.
$$42x^{2}-42x=7x\left(5x-3\right)$$
Use the distributive property to multiply $7x$ by $5x-3$.
$$42x^{2}-42x=35x^{2}-21x$$
Subtract $35x^{2}$ from both sides.
$$42x^{2}-42x-35x^{2}=-21x$$
Combine $42x^{2}$ and $-35x^{2}$ to get $7x^{2}$.
$$7x^{2}-42x=-21x$$
Add $21x$ to both sides.
$$7x^{2}-42x+21x=0$$
Combine $-42x$ and $21x$ to get $-21x$.
$$7x^{2}-21x=0$$
Factor out $x$.
$$x\left(7x-21\right)=0$$
To find equation solutions, solve $x=0$ and $7x-21=0$.
$$x=0$$ $$x=3$$
Variable $x$ cannot be equal to $0$.
$$x=3$$
Steps Using the Quadratic Formula
Variable $x$ cannot be equal to any of the values $0,\frac{3}{5}$ since division by zero is not defined. Multiply both sides of the equation by $42x\left(5x-3\right)$, the least common multiple of $7x\left(9x-\left(3+4x\right)\right),6$.
To find the opposite of $7-5x$, find the opposite of each term.
$$6x\left(2x-7+5x\right)=7x\left(5x-3\right)$$
Combine $2x$ and $5x$ to get $7x$.
$$6x\left(7x-7\right)=7x\left(5x-3\right)$$
Use the distributive property to multiply $6x$ by $7x-7$.
$$42x^{2}-42x=7x\left(5x-3\right)$$
Use the distributive property to multiply $7x$ by $5x-3$.
$$42x^{2}-42x=35x^{2}-21x$$
Subtract $35x^{2}$ from both sides.
$$42x^{2}-42x-35x^{2}=-21x$$
Combine $42x^{2}$ and $-35x^{2}$ to get $7x^{2}$.
$$7x^{2}-42x=-21x$$
Add $21x$ to both sides.
$$7x^{2}-42x+21x=0$$
Combine $-42x$ and $21x$ to get $-21x$.
$$7x^{2}-21x=0$$
This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $7$ for $a$, $-21$ for $b$, and $0$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.
Now solve the equation $x=\frac{21±21}{14}$ when $±$ is plus. Add $21$ to $21$.
$$x=\frac{42}{14}$$
Divide $42$ by $14$.
$$x=3$$
Now solve the equation $x=\frac{21±21}{14}$ when $±$ is minus. Subtract $21$ from $21$.
$$x=\frac{0}{14}$$
Divide $0$ by $14$.
$$x=0$$
The equation is now solved.
$$x=3$$ $$x=0$$
Variable $x$ cannot be equal to $0$.
$$x=3$$
Steps for Completing the Square
Variable $x$ cannot be equal to any of the values $0,\frac{3}{5}$ since division by zero is not defined. Multiply both sides of the equation by $42x\left(5x-3\right)$, the least common multiple of $7x\left(9x-\left(3+4x\right)\right),6$.
To find the opposite of $7-5x$, find the opposite of each term.
$$6x\left(2x-7+5x\right)=7x\left(5x-3\right)$$
Combine $2x$ and $5x$ to get $7x$.
$$6x\left(7x-7\right)=7x\left(5x-3\right)$$
Use the distributive property to multiply $6x$ by $7x-7$.
$$42x^{2}-42x=7x\left(5x-3\right)$$
Use the distributive property to multiply $7x$ by $5x-3$.
$$42x^{2}-42x=35x^{2}-21x$$
Subtract $35x^{2}$ from both sides.
$$42x^{2}-42x-35x^{2}=-21x$$
Combine $42x^{2}$ and $-35x^{2}$ to get $7x^{2}$.
$$7x^{2}-42x=-21x$$
Add $21x$ to both sides.
$$7x^{2}-42x+21x=0$$
Combine $-42x$ and $21x$ to get $-21x$.
$$7x^{2}-21x=0$$
Divide both sides by $7$.
$$\frac{7x^{2}-21x}{7}=\frac{0}{7}$$
Dividing by $7$ undoes the multiplication by $7$.
$$x^{2}+\left(-\frac{21}{7}\right)x=\frac{0}{7}$$
Divide $-21$ by $7$.
$$x^{2}-3x=\frac{0}{7}$$
Divide $0$ by $7$.
$$x^{2}-3x=0$$
Divide $-3$, the coefficient of the $x$ term, by $2$ to get $-\frac{3}{2}$. Then add the square of $-\frac{3}{2}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.
