Question

solve the equation by graphing the related function.
$x^2 + 3x + 4 = 0$
graph the related function. choose the correct answer below.
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Explanation:

Step1: Define related function

The related function is $y = x^2 + 3x + 4$.

Step2: Find vertex x-coordinate

Use vertex formula $x = -\frac{b}{2a}$. For $a=1, b=3$:
$x = -\frac{3}{2(1)} = -1.5$

Step3: Find vertex y-coordinate

Substitute $x=-1.5$ into the function:
$y = (-1.5)^2 + 3(-1.5) + 4 = 2.25 - 4.5 + 4 = 1.75$
Vertex is $(-1.5, 1.75)$.

Step4: Check y-intercept

Set $x=0$: $y=0+0+4=4$. So the graph crosses the y-axis at $(0,4)$.

Step5: Match to options

The parabola opens upward (since $a=1>0$), has vertex in the second quadrant, and y-intercept at $(0,4)$. This matches option B.

Step6: Solve the original equation

Since the graph of $y=x^2+3x+4$ never crosses the x-axis (vertex y-value is positive, parabola opens up), there are no real solutions.

Answer:

Correct graph: B. <The graph with upward-opening parabola, vertex in second quadrant, y-intercept at (0,4)>
Solution to $x^2 + 3x + 4 = 0$: No real solutions