Aim: What is the nature of the roots of a quadratic equation? Do Now: 1. Solve for x: 2. Solve for x: HW: p.202 #15,17,20,22,25,26,27

The roots for #1 are real, unequal and irrational numbers The roots for #2 are real, equal numbers. Look at another equation The roots are 7 and –1,which are real, unequal and rational numbers. . Let’s look at another equation the roots are They are imaginary numbers.

How do we determine what type of roots a certain equation has without solving it? We can use the discriminant to find out the nature of the roots. The discriminant is in the quadratic equation We use the value of b2 – 4ac to determine the properties of the roots of a quadratic equation.

There are 4 situations depending on the result of 1. b2 – 4ac > 0 and b2 – 4ac is a perfect square then the roots are real, unequal and rational numbers. 2. b2 – 4ac > 0 and b2 – 4ac is not a perfect square then the roots are real, unequal and irrational numbers. 3. b2 – 4ac = 0, then the roots are real, equal and rational numbers 4. b2 – 4ac < 0, then the roots are imaginary numbers

Find the discriminant then determine the nature of the equation 1. b2 – 4ac = 72 – 4(2)(-11) = 49 + 88 = 137 b2 – 4ac > 0, and 137 is not a perfect square, therefore the equation has real, unequal and irrational roots. 2. 3x2 – 4x + 5 = 0, b2 – 4ac = (-4)2 – 4(3)(5) = 16 – 60 = – 44 b2 – 4ac < 0, then the roots are imaginary

Example: Find the smallest possible integral value of c such that 3x2 – 7x + c = 0 will have imaginary roots: In order to have imaginary roots, the discriminant b2 – 4ac must be < 0. Therefore, c = 5

Determine k so that the equation has exactly one solution If the equation has exact one solution, then the discriminant b2 – 4ac = 0 302 – 4(9)(k) = 0 900 – 36k = 0 36k = 900, k = 25

Describe the roots of the following equations: 1. 2.. 3. 4. 5. Find the smallest possible integral value of c such that 6x2 – 4x + c = 0 will have imaginary roots. 6. Find the largest possible integral value of c such that 5x2 – 7x + c = 0 will have real roots

1. imaginary 2. real, unequal and rational 3. real, unequal and irrational 4. real and equal 5. 6. c = 2