3.3 Discriminants practice 1

Consider the quadratic function [math]f(x)=mx^2+4x+3[/math] where [math]m[/math] is a real number. If [math]f(x)[/math] is always positive when [math]a<m[/math], what is the value of [math]a[/math]?

Find the constant [math]k[/math] such that the equation [math]x^2=x+k[/math] has only one real solution.

The figure shows the graph of the quadratic function [math]f(x)=ax^2+bx+c[/math]. Identify the following signs. Identify the following signs. Write “[math]+[/math]”, “[math]-[/math]”, “[math]0[/math]” or “unknown”
(a) [math]a[/math]

(b) [math]b[/math]

(c) [math]c[/math]

(d) [math]b^2-4ac[/math]

(e) [math]a+b+c[/math]

(f) [math]a-b+c[/math]

If [math]f(x)=ax^2+bx+3[/math] and has exactly one zero at [math]x=\dfrac{1}{2}[/math]. If [math]a\neq0[/math], find [math]ab[/math].

For what is the maximum integer [math]k[/math] would the graph of [math]y=x^2-2x+k[/math] cut the [math]x[/math]-axis twice?

How many [math]x[/math]-intercepts does [math]f(x)=16x^2-8x+5[/math] have?

Which of the following equations has two complex, nonreal roots?

The graph of [math]f(x)=ax^2+bx+c[/math] is shown below.

(a) Find the value of [math]a[/math].

(b) The value of [math]b^2-4ac[/math] is: (write “[math]+[/math]”, “[math]-[/math]”, or “[math]0[/math]”)

(c) The roots of [math]f(x)[/math] are: (write “real” or “nonreal”)

(d) What is the maximum value of [math]f(x)[/math]?

(e) [math]f(6)=[/math]

How many [math]x[/math]-intercepts does [math]g(x)=-(x-30)^2+15[/math] have?

If [math]x^2+4x=3k[/math], what value of [math]k[/math] can make the equation has only one root?

For which of the following values of [math]a[/math] and [math]c[/math] does the equation [math]ax^2+3x+c=0[/math] have no real roots?

Given a quadratic function [math]f(x)=ax^2+bx+c[/math] with [math]a<0[/math] and two zeros at [math]x=-5[/math] and [math]x=3[/math], which of the following descriptions about [math]f(x)[/math] is correct?