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. $$ 2 x^{2}+5 x-7=0 $$

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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 $$ \boxed{B} $$. Option B has no real solutions.

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. \( 2 x^{2}+5 x-7=0 \)

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