To solve the equation $2x - x^2 = \frac{1}{4} x$, let's go through each step in detail:

Step 1: Set up the equation

The equation is: $2x - x^2 = \frac{1}{4} x$

Step 2: Move all terms to one side of the equation

Subtract $\frac{1}{4}x$ from both sides to get: $2x - x^2 - \frac{1}{4} x = 0$

Step 3: Combine like terms

Rewrite the equation, combining $2x$ and $-\frac{1}{4}x$: $\left(2 - \frac{1}{4}\right)x - x^2 = 0$

Calculating $2 - \frac{1}{4}$: $2 - \frac{1}{4} = \frac{8}{4} - \frac{1}{4} = \frac{7}{4}$

So the equation becomes: $\frac{7}{4}x - x^2 = 0$

Step 4: Rearrange the equation

Rewrite it in standard form: $-x^2 + \frac{7}{4}x = 0$

Or, multiplying through by $-1$ for simplicity: $x^2 - \frac{7}{4}x = 0$

Step 5: Factor the equation

Factor out $x$: $x \left(x - \frac{7}{4}\right) = 0$

Step 6: Solve for $x$

Set each factor equal to zero:

1. $x = 0$
2. $x - \frac{7}{4} = 0$

For the second equation, solve for $x$: $x = \frac{7}{4}$

Solution Summary

The two solutions are:

1. Smaller value: $x = 0$
2. Larger value: $x = \frac{7}{4}$

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Would you like further details on any of these steps or have any other questions?

Related Questions

1. How can we solve quadratic equations by factoring?
2. What are alternative methods to solve quadratic equations, like completing the square?
3. How does the distributive property assist in factoring equations?
4. How can we interpret the solutions to quadratic equations graphically?
5. What does it mean when an equation has only one solution?

Tip:

When solving quadratic equations, always check if it can be factored easily. Factoring can simplify finding solutions quickly without using the quadratic formula.