Use the discriminant to determine how many and what kind of solutions the quadratic equation x^2-x=-(1)/(4) one real solution two complex (nonreal) solutions no real or complex solutions two real solutions

## Step 1: Convert the equation to standard form<br />### Rewrite the given equation \( x^{2} - x = -\frac{1}{4} \) in standard quadratic form \( ax^2 + bx + c = 0 \). Add \(\frac{1}{4}\) to both sides to get \( x^2 - x + \frac{1}{4} = 0 \).<br />## Step 2: Identify coefficients<br />### Identify the coefficients from the standard form: \( a = 1 \), \( b = -1 \), and \( c = \frac{1}{4} \).<br />## Step 3: Calculate the discriminant<br />### Use the formula for the discriminant, \( \Delta = b^2 - 4ac \). Substitute the values: \( \Delta = (-1)^2 - 4(1)\left(\frac{1}{4}\right) \).<br />## Step 4: Simplify the discriminant<br />### Calculate \( \Delta = 1 - 1 = 0 \).<br />## Step 5: Determine the nature of the solutions<br />### Since the discriminant is zero (\(\Delta = 0\)), the quadratic equation has one real solution.