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Discriminants, Roots and Intercepts

Author:Nicholas Turner

Introduction

You are going to investigate the roots and x-intercepts of quadratic equations and graphs. The sliders a, b and c are the coefficients and constant of a quadratic equation The discriminant of a quadratic is the value of 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.

Select all that apply
  • A
  • B
  • C
  • D
  • E
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What can you say about the value of the discriminant when there are two x-intercepts?

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What can you say about the value of the discriminant when the graph touches the x-axis at exactly one point?

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What can you say about the value of the discriminant when the graph has no x-intercepts?

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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 has exactly one root. To answer the following questions, set

If , find the possible values of b so there is exactly one root.

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Factorise the quadratic expressions with these values of , and . What do you notice?

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Now set . Find the possible values of such that the graph touches the x-axis in exactly one point.

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Factorise the quadratic expression with these values of a, b and c. What do you notice?

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Repeat the previous questions with c = 9 and c = 16

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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.

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