SOLUTION: When an expression is real for all values of x, the why is it discriminant always less than 0. I was solving this question : The values of a for which the expression (ax^2 + 3x

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Question 1039247: When an expression is real for all values of x, the why is it discriminant always less than 0.
I was solving this question :
The values of a for which the expression (ax^2 + 3x -4 )/(3x - 4x^2 +a ) assumes all real values for real values of x, belongs to: ?
It would be more helpful to me if anyone can explain me the reason graphically?

Answer by josgarithmetic(39633) About Me  (Show Source):
You can put this solution on YOUR website!
Your numerator and your denominator both have roots or zeros; but remember that your rational expression will not accept its roots in its denominator.

An expression with no real roots will not intersect the x-axis.

Try solving a quadratic equation using Completing the Square. You will find an expression occurring called discriminant. Look at the lesson done all in symbols but no graph is present: Completing the Square to Solve Quadratic Equation -- https://www.algebra.com/my/Completing-the-Square-to-Solve-General-Quadratic-Equation.lesson?content_action=show_dev

Real solutions require the discriminant to be NON-ZERO; otherwise, solutions are complex with imaginary components; or, the roots are complex with imaginary components.