Solve (2m+5)/(2m-2)-(3-m)/(m)=(7)/(3) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{ 2m+5 }{ 2m-2 } - \frac{ 3-m }{ m } = \frac{ 7 }{ 3 }$$

Solve for m

$m=-2$
$m = \frac{9}{2} = 4\frac{1}{2} = 4.5$

Steps Using Factoring By Grouping

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using Factoring By Grouping

Variable $m$ cannot be equal to any of the values $0,1$ since division by zero is not defined. Multiply both sides of the equation by $6m\left(m-1\right)$, the least common multiple of $2m-2,m,3$.

$$3m\left(2m+5\right)-\left(6m-6\right)\left(3-m\right)=14m\left(m-1\right)$$

Use the distributive property to multiply $3m$ by $2m+5$.

$$6m^{2}+15m-\left(6m-6\right)\left(3-m\right)=14m\left(m-1\right)$$

Use the distributive property to multiply $6m-6$ by $3-m$ and combine like terms.

$$6m^{2}+15m-\left(24m-6m^{2}-18\right)=14m\left(m-1\right)$$

To find the opposite of $24m-6m^{2}-18$, find the opposite of each term.

$$6m^{2}+15m-24m+6m^{2}+18=14m\left(m-1\right)$$

Combine $15m$ and $-24m$ to get $-9m$.

$$6m^{2}-9m+6m^{2}+18=14m\left(m-1\right)$$

Combine $6m^{2}$ and $6m^{2}$ to get $12m^{2}$.

$$12m^{2}-9m+18=14m\left(m-1\right)$$

Use the distributive property to multiply $14m$ by $m-1$.

$$12m^{2}-9m+18=14m^{2}-14m$$

Subtract $14m^{2}$ from both sides.

$$12m^{2}-9m+18-14m^{2}=-14m$$

Combine $12m^{2}$ and $-14m^{2}$ to get $-2m^{2}$.

$$-2m^{2}-9m+18=-14m$$

Add $14m$ to both sides.

$$-2m^{2}-9m+18+14m=0$$

Combine $-9m$ and $14m$ to get $5m$.

$$-2m^{2}+5m+18=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as $-2m^{2}+am+bm+18$. To find $a$ and $b$, set up a system to be solved.

$$a+b=5$$ $$ab=-2\times 18=-36$$

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 $-36$.

$$-1,36$$ $$-2,18$$ $$-3,12$$ $$-4,9$$ $$-6,6$$

Calculate the sum for each pair.

$$-1+36=35$$ $$-2+18=16$$ $$-3+12=9$$ $$-4+9=5$$ $$-6+6=0$$

The solution is the pair that gives sum $5$.

$$a=9$$ $$b=-4$$

Rewrite $-2m^{2}+5m+18$ as $\left(-2m^{2}+9m\right)+\left(-4m+18\right)$.

$$\left(-2m^{2}+9m\right)+\left(-4m+18\right)$$

Factor out $-m$ in the first and $-2$ in the second group.

$$-m\left(2m-9\right)-2\left(2m-9\right)$$

Factor out common term $2m-9$ by using distributive property.

$$\left(2m-9\right)\left(-m-2\right)$$

To find equation solutions, solve $2m-9=0$ and $-m-2=0$.

$$m=\frac{9}{2}$$ $$m=-2$$

Steps Using the Quadratic Formula

Variable $m$ cannot be equal to any of the values $0,1$ since division by zero is not defined. Multiply both sides of the equation by $6m\left(m-1\right)$, the least common multiple of $2m-2,m,3$.

$$3m\left(2m+5\right)-\left(6m-6\right)\left(3-m\right)=14m\left(m-1\right)$$

Use the distributive property to multiply $3m$ by $2m+5$.

$$6m^{2}+15m-\left(6m-6\right)\left(3-m\right)=14m\left(m-1\right)$$

Use the distributive property to multiply $6m-6$ by $3-m$ and combine like terms.

$$6m^{2}+15m-\left(24m-6m^{2}-18\right)=14m\left(m-1\right)$$

To find the opposite of $24m-6m^{2}-18$, find the opposite of each term.

$$6m^{2}+15m-24m+6m^{2}+18=14m\left(m-1\right)$$

Combine $15m$ and $-24m$ to get $-9m$.

$$6m^{2}-9m+6m^{2}+18=14m\left(m-1\right)$$

Combine $6m^{2}$ and $6m^{2}$ to get $12m^{2}$.

$$12m^{2}-9m+18=14m\left(m-1\right)$$

Use the distributive property to multiply $14m$ by $m-1$.

$$12m^{2}-9m+18=14m^{2}-14m$$

Subtract $14m^{2}$ from both sides.

$$12m^{2}-9m+18-14m^{2}=-14m$$

Combine $12m^{2}$ and $-14m^{2}$ to get $-2m^{2}$.

$$-2m^{2}-9m+18=-14m$$

Add $14m$ to both sides.

$$-2m^{2}-9m+18+14m=0$$

Combine $-9m$ and $14m$ to get $5m$.

$$-2m^{2}+5m+18=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-2$ for $a$, $5$ for $b$, and $18$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$m=\frac{-5±\sqrt{5^{2}-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Square $5$.

$$m=\frac{-5±\sqrt{25-4\left(-2\right)\times 18}}{2\left(-2\right)}$$

Multiply $-4$ times $-2$.

$$m=\frac{-5±\sqrt{25+8\times 18}}{2\left(-2\right)}$$

Multiply $8$ times $18$.

$$m=\frac{-5±\sqrt{25+144}}{2\left(-2\right)}$$

Add $25$ to $144$.

$$m=\frac{-5±\sqrt{169}}{2\left(-2\right)}$$

Take the square root of $169$.

$$m=\frac{-5±13}{2\left(-2\right)}$$

Multiply $2$ times $-2$.

$$m=\frac{-5±13}{-4}$$

Now solve the equation $m=\frac{-5±13}{-4}$ when $±$ is plus. Add $-5$ to $13$.

$$m=\frac{8}{-4}$$

Divide $8$ by $-4$.

$$m=-2$$

Now solve the equation $m=\frac{-5±13}{-4}$ when $±$ is minus. Subtract $13$ from $-5$.

$$m=-\frac{18}{-4}$$

Reduce the fraction $\frac{-18}{-4}$ to lowest terms by extracting and canceling out $2$.

$$m=\frac{9}{2}$$

The equation is now solved.

$$m=-2$$ $$m=\frac{9}{2}$$

Steps for Completing the Square

Variable $m$ cannot be equal to any of the values $0,1$ since division by zero is not defined. Multiply both sides of the equation by $6m\left(m-1\right)$, the least common multiple of $2m-2,m,3$.

$$3m\left(2m+5\right)-\left(6m-6\right)\left(3-m\right)=14m\left(m-1\right)$$

Use the distributive property to multiply $3m$ by $2m+5$.

$$6m^{2}+15m-\left(6m-6\right)\left(3-m\right)=14m\left(m-1\right)$$

Use the distributive property to multiply $6m-6$ by $3-m$ and combine like terms.

$$6m^{2}+15m-\left(24m-6m^{2}-18\right)=14m\left(m-1\right)$$

To find the opposite of $24m-6m^{2}-18$, find the opposite of each term.

$$6m^{2}+15m-24m+6m^{2}+18=14m\left(m-1\right)$$

Combine $15m$ and $-24m$ to get $-9m$.

$$6m^{2}-9m+6m^{2}+18=14m\left(m-1\right)$$

Combine $6m^{2}$ and $6m^{2}$ to get $12m^{2}$.

$$12m^{2}-9m+18=14m\left(m-1\right)$$

Use the distributive property to multiply $14m$ by $m-1$.

$$12m^{2}-9m+18=14m^{2}-14m$$

Subtract $14m^{2}$ from both sides.

$$12m^{2}-9m+18-14m^{2}=-14m$$

Combine $12m^{2}$ and $-14m^{2}$ to get $-2m^{2}$.

$$-2m^{2}-9m+18=-14m$$

Add $14m$ to both sides.

$$-2m^{2}-9m+18+14m=0$$

Combine $-9m$ and $14m$ to get $5m$.

$$-2m^{2}+5m+18=0$$

Subtract $18$ from both sides. Anything subtracted from zero gives its negation.

$$-2m^{2}+5m=-18$$

Divide both sides by $-2$.

$$\frac{-2m^{2}+5m}{-2}=-\frac{18}{-2}$$

Dividing by $-2$ undoes the multiplication by $-2$.

$$m^{2}+\frac{5}{-2}m=-\frac{18}{-2}$$

Divide $5$ by $-2$.

$$m^{2}-\frac{5}{2}m=-\frac{18}{-2}$$

Divide $-18$ by $-2$.

$$m^{2}-\frac{5}{2}m=9$$

Divide $-\frac{5}{2}$, the coefficient of the $x$ term, by $2$ to get $-\frac{5}{4}$. Then add the square of $-\frac{5}{4}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{5}{2}m+\left(-\frac{5}{4}\right)^{2}=9+\left(-\frac{5}{4}\right)^{2}$$

Square $-\frac{5}{4}$ by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{5}{2}m+\frac{25}{16}=9+\frac{25}{16}$$

Add $9$ to $\frac{25}{16}$.

$$m^{2}-\frac{5}{2}m+\frac{25}{16}=\frac{169}{16}$$

Factor $m^{2}-\frac{5}{2}m+\frac{25}{16}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(m-\frac{5}{4}\right)^{2}=\frac{169}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{5}{4}\right)^{2}}=\sqrt{\frac{169}{16}}$$

Simplify.

$$m-\frac{5}{4}=\frac{13}{4}$$ $$m-\frac{5}{4}=-\frac{13}{4}$$

Add $\frac{5}{4}$ to both sides of the equation.

$$m=\frac{9}{2}$$ $$m=-2$$