Question

Assignment: The Discriminant

Answer 1

Answer:

1. D = 48, two; 2. D =  -8, none; 3. D =   0, one

4.D =  81, two; 5. D = -91, none; 6. D = 49, two

Step-by-step explanation:

The formula for a quadratic equation is

ax² + bx + c = 0

The quadratic formula gives the roots:

$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} = \dfrac{-b\pm\sqrt{D}}{2a}$

D is the discriminant.

It tells us the number of roots to the equation — the number of times the graph crosses the x-axis.

$D = \begin{cases}\text{positive} & \quad \text{2 real solutions}\\\text{zero} & \quad \text{1 real solution}\\\text{negative} & \quad \text{0 real solutions}\\\end{cases}$

It doesn't matter if the graph opens upwards or downwards.

If D > 0, the graph crosses the x-axis at two points.

If D = 0, the graph touches the x-axis at one point.

If D < 0, the graph never reaches the x-axis.

1. x² + 6x - 3 =0

a= 1; b = 6; c= -3

D = b² - 4ac  = 6² - 4×1×(-3) = 36 - 4(-3) = 36 + 12 = 48

D > 0: two real solutions

2. 3x² + 2x + 1 = 0

a = 3; b = 2; c =1

D = b² - 4ac  = 2² - 4×3×1 = 4 - 12 = -8

D < 0: no real solutions

3. x² + 4x + 4 = 0

a = 1; b = 4; c =1

D = 4² - 4×1×4  = 16 - 16 = 0

D = 0: one real solution

4. 5x² + x = 4

5x² + x - 4 = 0

a = 5; b = 4; c = -4

D = 1² - 4×5×(-4)  = 1 - 20(-4) = 1 - (-80) = 1 + 80  = 81

D > 0: two real solutions

5. 5x² - 3x = -5

5x² - 3x + 5=  0

a = 5; b = -3; c = 5

D = (-3)² - 4×5×5  = 9 - 100 =  -91

D < 0: no real solutions

6. x² - x  = 12

x² - x - 12=  0

a = 1; b = 1; c = -12

D = 1² - 4×1×(-12)  = 1 - 4(-12) = 1 + 48 =  49

D > 0: two real solutions