Name: ______Date: ______Class Period: _____

Nature of the Discriminant

Quadratic − b  b 2 − 4ac x = b2 − 4ac Discriminant Formula 2a

The discriminant predicts the “nature of the roots of a quadratic equation given that a, b, and c are rational numbers. It tells you the number of real roots/x-intercepts associated with a quadratic function.

Value of the Example showing nature of roots of Graph indicating x-intercepts Discriminant b2 – 4ac ax2 + bx + c = 0 for y = ax2 + bx + c POSITIVE Not a perfect x2 – 2x – 7 = 0 2 b – 4ac > 0 square − (−2)  (−2)2 − 4(1)(−7) x = 2(1) 2  32 2  4 2 x = = = 1 2 2 2 2 Discriminant: 32 There are two real roots. These roots are irrational. There are two x-intercepts. Perfect square x2 + 6x + 5 = 0 − 6  62 − 4(1)(5) x = 2(1) − 6  16 − 6  4 x = = = −1,−5 2 2 Discriminant: 16 There are two real roots. These roots are rational. There are two x-intercepts. ZERO b2 – 4ac = 0 x2 – 2x + 1 = 0 − (−2)  (−2)2 − 4(1)(1) x = 2(1) 2  0 2 x = = = 1 2 2 Discriminant: 0 There is one real root (with a multiplicity of 2). This root is rational. There is one x-intercept. NEGATIVE b2 – 4ac < 0 x2 – 3x + 10 = 0 − (−3)  (−3)2 − 4(1)(10) x = 2(1)

3  − 31 3 31 x = =  i 2 2 2 Discriminant: -31 There are two complex/imaginary roots. There are no x-intercepts.

Quadratic Formula and Discriminant Practice

1. Which equation has imaginary roots? 7. How many x-intercepts does the graph of the 2 a. x2 – 9 = 0 function f(x) = x – 4x + 11 = 0 have? 2 b. x – x + 1 = 0 a. None 2 c. x + 2x + 1 = 0 b. One 2 d. x – 1 = 0 c. Two 2. What is the nature of the roots of 4x2 – 4x + 1 = 0? d. Cannot be determined a. Two real, rational solutions 8. Does the vertex of the graph of the function f(x) = b. Two real, irrational solutions x2 – 4x + 11 = 0 lie above, below, or on the x-axis? c. One real, rational solution a. Above d. Two imaginary/complex solutions b. Below 3. How many x-intercepts does the graph of the c. On function f(x) = x2 – 7x + 7 = 0 have? d. Cannot be determined a. None 9. How many solutions does 2x2 – 6x + 10 = 0 have? b. One c. Two d. Cannot be determined

4. Does the vertex of the graph of the function f(x) = 10. How many solutions does 3x2 – 6x – 10 = 0 have? x2 – 7x + 7 = 0 lie above, below, or on the x-axis? a. Above b. Below c. On 11. How many solutions does the function d. Cannot be determined x2 – 2x + 1 = 0 have? 5. How many x-intercepts does the graph of the function f(x) = x2 – 3x + 1 = 0 have? a. None b. One 12. Find all k such that 2x2 + kx + 3 = 0 has only one c. Two solution? Hint: the discriminant is k2 – 24; solve d. Cannot be determined k2 – 24 = 0. 6. Does the vertex of the graph of the function f(x) = x2 – 3x + 1 = 0 lie above, below, or on the x-axis? a. Above b. Below 13. Find all k such that x2 + kx + 2k = 0 has only one c. On solution? d. Cannot be determined