4. Which equation has no real solutions? A. \( x^{2}-8 x+16=0 \) B. \( x^{2}-3 x+4=0 \) C. \( 2 x^{2}-11 x+5=0 \) D. \( 2 x^{2}+5 x-7=0 \)

1. For a quadratic equation of the form \( ax^2+bx+c=0 \), the discriminant is given by \[ D = b^2 - 4ac. \] The equation has no real solutions if \( D < 0 \). 2. Consider each equation: **Option A:** The equation is \( x^2 - 8x + 16 = 0 \) with \( a = 1 \), \( b = -8 \), and \( c = 16 \). Calculate the discriminant: \[ D = (-8)^2 - 4(1)(16) = 64 - 64 = 0. \] Since \( D = 0 \), there is one repeated real solution. **Option B:** The equation is \( x^2 - 3x + 4 = 0 \) with \( a = 1 \), \( b = -3 \), and \( c = 4 \). Calculate the discriminant: \[ D = (-3)^2 - 4(1)(4) = 9 - 16 = -7. \] Since \( D < 0 \), this equation has no real solutions. **Option C:** The equation is \( 2x^2 - 11x + 5 = 0 \) with \( a = 2 \), \( b = -11 \), and \( c = 5 \). Calculate the discriminant: \[ D = (-11)^2 - 4(2)(5) = 121 - 40 = 81. \] Since \( D > 0 \), there are two distinct real solutions. **Option D:** The equation is \( 2x^2 + 5x - 7 = 0 \) with \( a = 2 \), \( b = 5 \), and \( c = -7 \). Calculate the discriminant: \[ D = (5)^2 - 4(2)(-7) = 25 + 56 = 81. \] Since \( D > 0 \), there are two distinct real solutions. 3. From the discriminant analysis, only Option B has a negative discriminant. Thus, the equation with no real solutions is Option \( \boxed{B} \).