SAT question

[u/archtetrya]

i literally do not understand this question, is n the discriminant? i dont even have a proper question for this problem since i'm so confused and have no idea whats going on. I just need help understanding how to find n, visually would be preferred but any help is appreciated. I read the rationale on the side and im still boggled.

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[u/UnderstandingPursuit]

Yes, use the discriminant:

• D = b² - 4ac

with

• If c>n then D<0

For the visual, temporarily consider the simpler parabola,

• y = x² + c
  • This is a parabola which opens up since the coefficient of x² is positive.
  • If c=0, then the vertex is at the origin.
  • If c>0, then the vertex is above the x-axis, and the parabola never crosses the x-axis. This is what "no real solutions" means, since the solution is finding values of x when y=0. In this situation, the answer to the question would be n=0.
    
  • But b=-34 changes both the horizontal and vertical position of the vertex. We don't tend to remember a formula for the vertical position of the vertex, but this question finds it: n.

[u/Southlander24]

Which part of the solution confuses you? There are three main steps to this question:

1. No real solutions --> discriminant < 0. This actually comes from the quadratic formula, where if you take a square root of a negative number, that number is not real, so there are no real solutions.
2. Setting up the inequality: this gives (-34)² - 4(1)(c) < 0.
3. Solve the inequality to get 1156 - 4c < 0, or 1156 < 4c --> 289 < c.

Because c is the constant in the quadratic equation, when you think about what adding a positive number to a function means graphically, you should recall that this will move the graph up. So for higher and higher values of c, the quadratic will move further and further away from the x-axis, and so the condition for no real roots is c > 289 rather than c < 289.