3.3 Discriminants (Part 2)

Let [math]x[/math] be a real number. If [math]k=\dfrac{x^2-x+1}{x^2+x+1}[/math] , what is the maximum of [math]k[/math] ?

Hint: Rewrite the equation as a quadratic equation and apply the discriminant

If [math]a<x<b[/math] is the solution to the inequality [math]-(1-2x)(3-x)>0[/math], what is the value of [math]a+b[/math]?

Which of the following is the solution to the inequality [math]x^2-8x\leq16[/math]?

How many integer solutions does the inequality [math]x^2<49[/math] have?

Which of the following is the solution to the inequality [math]x^2-8x+12>0[/math]?

Let [math]p, q[/math] be the real solutions of the quadratic equation [math]x^2-(k+2)x+(k^2+k-2)[/math] where [math]k\in\mathbb{R}[/math]. Find the maximum of [math]p^2+q^2[/math].

Hint: sum and product of roots

If [math]m>a[/math] where [math]a[/math] is a real number, then [math]9(m-1)x^2-6mx+(m+3)[/math] is always positive. Find the value of [math]a[/math].

How many integer solutions does the equation [math]-2x^2+5x+3\geq0[/math] have?

How many integer solutions does the equation [math]x^2\leq9x[/math] have?

If [math]a<k<b[/math], then the equation [math]2x^2+(3-k)x+k+3=0[/math] has 2 imaginary solutions, what is the value of [math]a+b[/math]?

If [math]a<p<b[/math], then [math]f(x)=x^3+3px+p[/math] is greater than zero for all real values of [math]x[/math]. What is the value of [math]a[/math]?