Questions: Use the discriminant to determine the number of real solutions. w^2-2w+2=0 No real solutions One real solution Two real solutions

Solution Steps

To determine the number of real solutions for the quadratic equation \( w^2 - 2w + 2 = 0 \), we use the discriminant. The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). The number of real solutions depends on the value of the discriminant:

• If \( \Delta > 0 \), there are two real solutions.
• If \( \Delta = 0 \), there is one real solution.
• If \( \Delta < 0 \), there are no real solutions.

For the given equation \( w^2 - 2w + 2 = 0 \):

• \( a = 1 \)
• \( b = -2 \)
• \( c = 2 \)

We will calculate the discriminant and determine the number of real solutions.

For the quadratic equation \( w^2 - 2w + 2 = 0 \), we identify the coefficients as follows:

• \( a = 1 \)
• \( b = -2 \)
• \( c = 2 \)

The discriminant \( \Delta \) is calculated using the formula: \[ \Delta = b^2 - 4ac \] Substituting the values of \( a \), \( b \), and \( c \): \[ \Delta = (-2)^2 - 4 \cdot 1 \cdot 2 = 4 - 8 = -4 \]

Since the discriminant \( \Delta = -4 \) is less than zero (\( \Delta < 0 \)), this indicates that there are no real solutions to the equation.

Final Answer