4E - The discriminant
The discriminant (\(\Delta\)) comes from the expression under the square root sign in the general quadratic formula.
■ For a quadratic equation, \(ax^2+bx+c=0\), the discriminant is defined to be \[\Delta = b^2-4ac\] ■ If the value of \(\Delta\) is a square number (\(1,4,9,16,...\)) then the solutions to the equation will be rational.
By considering the value of \(\Delta\), we can determine how many solutions (or \(x\)-intercepts) there are:
- If \(\Delta>0\) (positive), two solutions exist. We can interpret this as 2 \(x\)-intercepts.
- If \(\Delta=0\) (zero), one solution exists. We can interpret this as 1 \(x\)-intercepts.
- If \(\Delta<0\) (negative), no solutions exist. We can interpret this as 0 \(x\)-intercepts.
4E - Example 1: Determining the number of solutions using the discriminant
Use the discriminant to determine how many solutions exist for each of the following:
(a) \(x^2-2x+5=0\) (b) \(x^2-6x+9=0\) (c) \(x^2+2x-15=0\) |
4E - Example 1: Video solution
4E - Example 1: Practice
Question 1: ABC 4E - Example 1: Solutions
Question 1: ABC |
4E - Example 2: Applying the discriminant to show that solutions are rational or irrational
Use the discriminant to show that the quadratic \[y=-x^2+(m+2)x+3-3m\] has rational \(x\)-intercepts for any value of \(m\in \mathbb{Z}\).
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4E - Example 2: Video solution
4E - Example 2: Practice
Question 1: Find the value(s) of \(m\) such that the graph of \(y=x^2+mx+5\) has (a) No \(x\)-intercepts. (b) One \(x\)-intercept. (c) Two \(x\)-intercepts. (d) Two rational \(x\)-intercepts. 4E - Example 2: Solutions
Question 1: (a) \(-2\sqrt{5}\leq m \leq 2\sqrt{5}\) (b) \(m=-2\sqrt{5}\) or \(m=2\sqrt{5}\). (c) \(m<-2\sqrt{5}\) or \(m>2\sqrt{5}\) (d) Many possible if \(\Delta\) is a square number \[m^2-20=16\therefore m=\pm 6\] |
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Additional Exercises
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Topic Worksheets
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Other Resources
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The general quadratic formula and the discriminant - Worksheet A
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