You are going to investigate the roots and x-intercepts of quadratic equations and graphs.[br]The sliders a, b and c are the coefficients and constant of a quadratic equation [math]y=ax^2+bx+c[/math][br][br]The discriminant of a quadratic is the value of [math]b^2-4ac[/math][br][br]Use the applet below to answer the questions.
Which of the following are possible numbers of x-intercepts for the graph of a quadratic equation.
What can you say about the value of the discriminant when there are two x-intercepts?
It is positive
What can you say about the value of the discriminant when the graph touches the x-axis at exactly one point?
It is equal to zero
What can you say about the value of the discriminant when the graph has no x-intercepts?
It is negative
For the next part of this activity you are going to be investigating the special cases where the graph touches the x-axis at exactly one point. This is when the vertex of graph is on the x-axis. We also say that the equation [math]ax^2+bx+c=0[/math] has exactly one root.[br]To answer the following questions, set [math]a=1[/math]
If [math]c=1[/math], find the possible values of b so there is exactly one root.
[math]b=2[/math] or [math]b=-2[/math]
Factorise the quadratic expressions with these values of [math]a[/math], [math]b[/math] and [math]c[/math]. What do you notice?
[math]\left(x-1\right)^2[/math] and [math]\left(x+1\right)^2[/math][br][br]They are perfect squares.
Now set [math]c=4[/math]. Find the possible values of [math]b[/math] such that the graph touches the x-axis in exactly one point.
[math]b=-4[/math] and [math]b=4[/math][br]
Factorise the quadratic expression with these values of a, b and c. What do you notice?
[math]\left(x-2\right)^2[/math] and [math]\left(x+2\right)^2[/math] they are perfect squares
Repeat the previous questions with c = 9 and c = 16
For c = 9, b = -6 or b = 6 Factorised expression [math]\left(x-3\right)^2[/math] or [math]\left(x+3\right)^2[/math][br][br]For c = 16, b = -8 or b = 8 Factorised expression [math]\left(x-4\right)^2[/math] or [math]\left(x+4\right)^2[/math]
Generalising your results
From what you have found out from the previous questions, write an equation relating the value of b to the value of c, for a quadratic graph with exactly one root.