Quadratics  the Discriminant.
Test Yourself 1.
To answer the following questions  you will need to use the discriminant:
 to calculate the value of the discriminant;
 to interpret the value directly;
 to find values.
No real roots.  1. Show that the equation x^{2} + 5x + 8 = 0 has no real roots. 
2. Show that the equation 2x^{2}  12x + 25 = 0 has no real roots. 
3. For what values of p will the quadratic equation
x^{2} + (p  2)x + 1 = 0 have unreal roots? 
4. Explain why the graph of the parabola described by the equation
y = 3x^{2}  2x + 4 is located above the x axis for all values of x. 

Real roots  5.For what value of m are the roots of f(x) = mx^{2} + 6x  3 real?

6. (i) Find the discriminant of the equation
3x^{2} + 2k + k = 0.
(ii) For what values of k does the equation

7. Show that x^{2} + kx + k  1 = 0 has real roots for all values of k.  8.  
One real root  9. Find the value(s) of k for which the quadratic equation
3x^{2} + 2x + k = 0 has one real root. Answer. k = 1/3. 
10. Find the values of k for which the quadratic equation x^{2}  (k + 4)x + (k + 7) = 0 has equal roots. Answer. k = 4 or k = 8. 
11. For what value(s) of k does the equation y = x^{2}  2kx + 6k have one real root? Answer. k = 0 or k = 6. 
12.


Real and different roots.  13. Find the value(s) of k if the roots of the equation x^{2} + kx + 36 = 0 are real and distinct. 
14. 
15. For the quadratic equation
x^{2} + (p  3)x  (2p + 1) = 0

16.  
Mixed questions.  17. For what values of k does
x^{2}  (k + 5)x + 9 = 0 have


Positive definite.  19. For what values of k is the equation kx^{2}  (k + 1)  2k + 3 = 0 positive definite?  20. For what values of k is the equation
(k + 1) x^{2}  2(2 + k) + 3 = 0 positive definite? Answer. 4  2√6 < k < 4 + 2√6. 
21.  
Negative definite.  23. 

25.  26.  
Advanced contexts.  27. (i) Show that for all values of m, the line y = mx  3m^{2} touches the parabola x^{2} = 12y.
(ii) Find the values of m for which this line passes through the point (5,2). (iii) Hence determine the equations of the two tangents to the given parabola from the point (5,2). 


28. The circle x^{2} + (y – c)^{2} = r^{2}, where c > 0 and r > 0, lies inside the parabola y = x^{2}. The circle touches the parabola at exactly two points located symmetrically on opposite sides of the yaxis, as shown in the diagram.


29. The parabola y = b  ax^{2} , where a > 0 and b > 0 , is under the curve .
The parabola touches the curve at two points that are symmetrical at the yaxis, as shown in the diagram above.


30. (i) Write down the discriminant of 2x^{2} + (k –2)x + 8, where k is a constant.
(ii) Hence, or otherwise, find the values of k for which the parabola


Increasing functions.  31. For what values of k is the curve y = x^{3}  3x^{2} + kx + 3 always an increasing function?  
32.  
Decreasing functions.  33.  
34. 