2. how many real solutions and how many non-real solutions does the equation $x^2+5=0$ have? describe what implications your answer has for graphing the function $y=x^2+5$.

3. solve the quadratic equation using your choice of the methods we learned in this module. indicate which method you are using, and show your work.
\4x^2 - 22 = x^2 + 5\

Question

2. how many real solutions and how many non-real solutions does the equation $x^2+5=0$ have? describe what implications your answer has for graphing the function $y=x^2+5$.

3. solve the quadratic equation using your choice of the methods we learned in this module. indicate which method you are using, and show your work.
\4x^2 - 22 = x^2 + 5\

Accepted Answer

{ "quality": "clear", "question_count": 2, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [], "new_concepts": [ "Quadratic Discriminant", "Complex Roots", "Parabola Graphing" ], "current_concepts": [ "Quadratic Equations", "Real and Complex Roots", "Graphical Interpretation" ] } ## Solve the first equation and determine root types $$ \begin{aligned} &x^2 + 5 = 0 \ &x^2 = -5 \ &x = \pm i\sqrt{5} \end{aligned} $$ The equation has $0$ real solutions and $2$ non-real (complex) solutions. Since there are no real solutions, the graph of $y = x^2 + 5$ has no $x$-intercepts and does not cross or touch the $x$-axis. ## Solve the second quadratic equation $$ \begin{aligned} &4x^2 - 22 = x^2 + 5 \ &3x^2 = 27 \ &x^2 = 9 \ &x = \pm 3 \end{aligned} $$ The method used is solving by taking square roots. ## Question 2 The equation $x^2+5=0$ has $0$ real solutions and $2$ non-real solutions. This implies that the graph of the function $y=x^2+5$ has no $x$-intercepts and does not intersect the $x$-axis. ## Question 3 Method: Solving by taking square roots. $$ \begin{aligned} 4x^2 - 22 &= x^2 + 5 \ 3x^2 &= 27 \ x^2 &= 9 \ x &= \pm 3 \end{aligned} $$ The solutions are $x = 3$ and $x = -3$. { "elements": [ { "type": "functiongraph", "params": [ { "js": "x*x + 5", "latex": "x^2 + 5" }, -4, 4 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = x^2 + 5", "withLabel": true } }, { "type": "line", "params": [ [-10, 0], [10, 0] ], "properties": { "strokeColor": "#583C87", "strokeWidth": 1, "dash": 1 } } ], "timestamps": [0.5] } { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Quadratic Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Solve the first equation and determine root types

Solve the second quadratic equation

Answer:

Question 2

Question 3