Questions: Determine the discriminant and show how many real solutions there are 2 x^2-6 x+7=0

Solution Steps

To determine the discriminant of a quadratic equation and find out how many real solutions it has, we use the formula for the discriminant, \( D = b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). The number of real solutions is determined by the value of the discriminant: if \( D > 0 \), there are two distinct real solutions; if \( D = 0 \), there is one real solution; and if \( D < 0 \), there are no real solutions.

For the quadratic equation \( 2x^2 - 6x + 7 = 0 \), the coefficients are:

• \( a = 2 \)
• \( b = -6 \)
• \( c = 7 \)

The discriminant \( D \) is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values: \[ D = (-6)^2 - 4 \cdot 2 \cdot 7 = 36 - 56 = -20 \]

Since the discriminant \( D = -20 \) is less than zero, this indicates that there are no real solutions to the equation.

Final Answer