which of the following has no real solution?

$x^2 - 4x - 5 = 0$

$x^2 - 10x + 3 = 0$

$x^2 - 4 = 0$

$x^2 + 2x + 2 = 0$

Question

which of the following has no real solution?

$x^2 - 4x - 5 = 0$

$x^2 - 10x + 3 = 0$

$x^2 - 4 = 0$

$x^2 + 2x + 2 = 0$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Quadratic Equations", "Quadratic Formula" ], "new_concepts": [ "Discriminant Analysis" ], "current_concepts": [ "Quadratic Equations", "Quadratic Formula", "Discriminant Analysis" ] } ## Define the discriminant of a quadratic equation $$ D = b^2 - 4ac $$ ## Calculate the discriminant for each option $$ \begin{aligned} &\text{Option 1: } x^2 - 4x - 5 = 0 \implies D = (-4)^2 - 4(1)(-5) = 16 + 20 = 36 > 0 \ &\text{Option 2: } x^2 - 10x + 3 = 0 \implies D = (-10)^2 - 4(1)(3) = 100 - 12 = 88 > 0 \ &\text{Option 3: } x^2 - 4 = 0 \implies D = 0^2 - 4(1)(-4) = 16 > 0 \ &\text{Option 4: } x^2 + 2x + 2 = 0 \implies D = 2^2 - 4(1)(2) = 4 - 8 = -4 < 0 \end{aligned} $$ ## Identify the equation with no real solution $$ D < 0 \implies x^2 + 2x + 2 = 0 $$

(A) $x^2 - 4x - 5 = 0$

(B) $x^2 - 10x + 3 = 0$

(C) $x^2 - 4 = 0$

(D) $x^2 + 2x + 2 = 0$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Define the discriminant of a quadratic equation

Calculate the discriminant for each option

Identify the equation with no real solution

Answer: