How to Solve for q?

Solving for a variable like $q$ involves isolating it on one side of the equation. Let’s go through a few common types of equations to understand how to solve for $q$ in each scenario.

Linear Equations

A linear equation is of the form $aq + b = c$. To solve for $q$, follow these steps:

1. Subtract $b$ from both sides: $aq = c – b$
2. Divide by $a$: $q = frac{c – b}{a}$

Example

Given the equation $3q + 5 = 11$:

1. Subtract 5 from both sides: $3q = 6$
2. Divide by 3: $q = frac{6}{3} = 2$

Quadratic Equations

A quadratic equation is of the form $aq^2 + bq + c = 0$. To solve for $q$, you can use the quadratic formula:

$q = frac{-b , pm , sqrt{b^2 – 4ac}}{2a}$

Example

Given $q^2 – 4q + 4 = 0$:

1. Identify $a$, $b$, and $c$: $a = 1$, $b = -4$, $c = 4$
2. Plug into the quadratic formula:

$q = frac{-(-4) , pm , sqrt{(-4)^2 – 4 cdot 1 cdot 4}}{2 cdot 1}$

$q = frac{4 , pm , sqrt{16 – 16}}{2}$

$q = frac{4 , pm , 0}{2}$

$q = 2$

Exponential Equations

An exponential equation is of the form $a^q = b$. To solve for $q$, take the logarithm of both sides:

$q = log_a{b}$

Example

Given $2^q = 16$:

1. Take the logarithm base 2 of both sides: $q = log_2{16}$
2. Since $16 = 2^4$, $q = 4$

Logarithmic Equations

A logarithmic equation is of the form $log_a{q} = b$. To solve for $q$, rewrite the equation in its exponential form:

$q = a^b$

Example

Given $log_2{q} = 3$:

1. Rewrite in exponential form: $q = 2^3$
2. Calculate $q$: $q = 8$

Conclusion

Solving for $q$ depends on the type of equation you’re dealing with. Whether it’s linear, quadratic, exponential, or logarithmic, the key is to isolate $q$ using algebraic operations or specific formulas. Practice with different types of equations to become more comfortable with these methods.

Citations

1. 1. Khan Academy – Solving Equations
2. 2. Math is Fun – Solving Equations
3. 3. Purplemath – Solving Equations