$$4 p \frac { 5 p - 3 } { 2 p } = \frac { 8 } { 9 }$$
Solve for p
$p=\frac{31}{45}\approx 0.688888889$
Steps Using Factoring
Steps Using the Quadratic Formula
Steps for Completing the Square
Steps Using Factoring
Variable $p$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $18p$, the least common multiple of $2p,9$.
$$4p\times 9\left(5p-3\right)=16p$$
Multiply $4$ and $9$ to get $36$.
$$36p\left(5p-3\right)=16p$$
Use the distributive property to multiply $36p$ by $5p-3$.
$$180p^{2}-108p=16p$$
Subtract $16p$ from both sides.
$$180p^{2}-108p-16p=0$$
Combine $-108p$ and $-16p$ to get $-124p$.
$$180p^{2}-124p=0$$
Factor out $p$.
$$p\left(180p-124\right)=0$$
To find equation solutions, solve $p=0$ and $180p-124=0$.
$$p=0$$ $$p=\frac{31}{45}$$
Variable $p$ cannot be equal to $0$.
$$p=\frac{31}{45}$$
Steps Using the Quadratic Formula
Variable $p$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $18p$, the least common multiple of $2p,9$.
$$4p\times 9\left(5p-3\right)=16p$$
Multiply $4$ and $9$ to get $36$.
$$36p\left(5p-3\right)=16p$$
Use the distributive property to multiply $36p$ by $5p-3$.
$$180p^{2}-108p=16p$$
Subtract $16p$ from both sides.
$$180p^{2}-108p-16p=0$$
Combine $-108p$ and $-16p$ to get $-124p$.
$$180p^{2}-124p=0$$
This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $180$ for $a$, $-124$ for $b$, and $0$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.
Now solve the equation $p=\frac{124±124}{360}$ when $±$ is plus. Add $124$ to $124$.
$$p=\frac{248}{360}$$
Reduce the fraction $\frac{248}{360}$ to lowest terms by extracting and canceling out $8$.
$$p=\frac{31}{45}$$
Now solve the equation $p=\frac{124±124}{360}$ when $±$ is minus. Subtract $124$ from $124$.
$$p=\frac{0}{360}$$
Divide $0$ by $360$.
$$p=0$$
The equation is now solved.
$$p=\frac{31}{45}$$ $$p=0$$
Variable $p$ cannot be equal to $0$.
$$p=\frac{31}{45}$$
Steps for Completing the Square
Variable $p$ cannot be equal to $0$ since division by zero is not defined. Multiply both sides of the equation by $18p$, the least common multiple of $2p,9$.
$$4p\times 9\left(5p-3\right)=16p$$
Multiply $4$ and $9$ to get $36$.
$$36p\left(5p-3\right)=16p$$
Use the distributive property to multiply $36p$ by $5p-3$.
$$180p^{2}-108p=16p$$
Subtract $16p$ from both sides.
$$180p^{2}-108p-16p=0$$
Combine $-108p$ and $-16p$ to get $-124p$.
$$180p^{2}-124p=0$$
Divide both sides by $180$.
$$\frac{180p^{2}-124p}{180}=\frac{0}{180}$$
Dividing by $180$ undoes the multiplication by $180$.
Reduce the fraction $\frac{-124}{180}$ to lowest terms by extracting and canceling out $4$.
$$p^{2}-\frac{31}{45}p=\frac{0}{180}$$
Divide $0$ by $180$.
$$p^{2}-\frac{31}{45}p=0$$
Divide $-\frac{31}{45}$, the coefficient of the $x$ term, by $2$ to get $-\frac{31}{90}$. Then add the square of $-\frac{31}{90}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.
Factor $p^{2}-\frac{31}{45}p+\frac{961}{8100}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.
