Solve 1/4 * q^2 = q - 1
Understand the Problem
The question asks us to solve for q in the equation 1/4 * q^2 = q - 1. This involves manipulating the equation, potentially rearranging it into a standard quadratic form, and then applying techniques like factoring, completing the square, or using the quadratic formula to find q.
Answer
$q = 2$
Answer for screen readers
$q = 2$
Steps to Solve
- Multiply both sides by 4 to eliminate the fraction
To get rid of the fraction, multiply both sides of the equation by 4.
$$ 4 * (\frac{1}{4}q^2) = 4 * (q - 1) $$
This simplifies to:
$$ q^2 = 4q - 4 $$
- Rearrange the equation into standard quadratic form
Subtract $4q$ and add $4$ to both sides to set the equation equal to zero, resulting in the standard quadratic form $aq^2 + bq + c = 0$.
$$ q^2 - 4q + 4 = 0 $$
- Factor the quadratic equation
Now, factor the quadratic equation. We are looking for two numbers that multiply to 4 and add to -4. Those numbers are -2 and -2.
$$ (q - 2)(q - 2) = 0 $$
Or
$$ (q - 2)^2 = 0 $$
- Solve for q
Set each factor equal to zero and solve for $q$. In this case, both factors are the same.
$$ q - 2 = 0 $$
Add 2 to both sides:
$$ q = 2 $$
$q = 2$
More Information
The equation has only one solution, $q = 2$. This occurs because the quadratic expression is a perfect square.
Tips
A common mistake is to incorrectly factor the quadratic equation or to make an error when applying the quadratic formula. Also, some might forget to rearrange the equation into standard quadratic form before attempting to factor or use the quadratic formula.
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