Question

1. match the following words to each graph:

a. 1 real solution \t\ti. discriminant is positive
b. 2 real solutions \t\tii. discriminant is negative
c. 2 complex solutions \tiii. discriminant is zero
this graph has __ & \tthis graph has & \tthis graph has & __
(graphs of parabolas are shown here)

1. simplify each of the following complex expressions.

a. $(3 + 2i) + (4 - 6i)$ \t b. $(3 + 2i) - (4 - 6i)$ \t c. $(2 + i) + (5 - 4i)$
d. $(2 + i) - (5 - 4i)$ \t e. $(3 + 2i)(4 - 6i)$ \t f. $(2 + i)(5 - 4i)$
g. $i(- 3 + 2i)$ \t\t h. $5i(2 - 4i)$ \t\t i. $(4i)(2i)(3 - 4i)$

Explanation:

Part 5 (Matching Graphs and Solutions/Discriminants)

First Graph (Parabola intersecting x - axis at two points)

Step1: Analyze the number of real solutions

A parabola that intersects the x - axis at two distinct points has 2 real solutions. So it matches with option B.

Step2: Analyze the discriminant

For a quadratic equation \(y = ax^{2}+bx + c\), the discriminant is \(D=b^{2}-4ac\). If there are 2 real solutions, the discriminant is positive (since \(x=\frac{-b\pm\sqrt{D}}{2a}\), and if \(D>0\), we have two distinct real roots). So it matches with I.

Second Graph (Parabola not intersecting x - axis)

Step1: Analyze the number of real solutions

A parabola that does not intersect the x - axis has no real solutions, which means it has 2 complex solutions (since the solutions are of the form \(x=\frac{-b\pm\sqrt{D}}{2a}\), and if \(D < 0\), \(\sqrt{D}\) is imaginary). So it matches with option C.

Step2: Analyze the discriminant

If there are no real solutions (2 complex solutions), the discriminant is negative. So it matches with II.

Third Graph (Parabola tangent to x - axis at one point)

Step1: Analyze the number of real solutions

A parabola that is tangent to the x - axis at one point has 1 real solution (a repeated root). So it matches with option A.

Step2: Analyze the discriminant

If there is 1 real solution (a repeated root), the discriminant is zero (since \(x=\frac{-b\pm\sqrt{D}}{2a}\), and if \(D = 0\), we have one real root \(\frac{-b}{2a}\)). So it matches with III.

Part 6 (Simplifying Complex Expressions)

a. \((3 + 2i)+(4 - 6i)\)

Step1: Combine real parts and imaginary parts

Real parts: \(3 + 4=7\)
Imaginary parts: \(2i-6i=-4i\)
So \((3 + 2i)+(4 - 6i)=7 - 4i\)

b. \((3 + 2i)-(4 - 6i)\)

Step1: Distribute the negative sign

\(3 + 2i-4 + 6i\)

Step2: Combine real parts and imaginary parts

Real parts: \(3-4=-1\)
Imaginary parts: \(2i + 6i=8i\)
So \((3 + 2i)-(4 - 6i)=-1 + 8i\)

c. \((2 + i)+(5 - 4i)\)

Step1: Combine real parts and imaginary parts

Real parts: \(2 + 5=7\)
Imaginary parts: \(i-4i=-3i\)
So \((2 + i)+(5 - 4i)=7 - 3i\)

d. \((2 + i)-(5 - 4i)\)

Step1: Distribute the negative sign

\(2 + i-5 + 4i\)

Step2: Combine real parts and imaginary parts

Real parts: \(2-5=-3\)
Imaginary parts: \(i + 4i=5i\)
So \((2 + i)-(5 - 4i)=-3 + 5i\)

e. \((3 + 2i)(4 - 6i)\)

Step1: Use the FOIL method

\(3\times4+3\times(-6i)+2i\times4+2i\times(-6i)\)
\(=12-18i + 8i-12i^{2}\)

Step2: Recall that \(i^{2}=-1\)

\(=12-10i-12\times(-1)\)
\(=12-10i + 12\)

Step3: Combine like terms

\(=24-10i\)

f. \((2 + i)(5 - 4i)\)

Step1: Use the FOIL method

\(2\times5+2\times(-4i)+i\times5+i\times(-4i)\)
\(=10-8i + 5i-4i^{2}\)

Step2: Recall that \(i^{2}=-1\)

\(=10-3i-4\times(-1)\)
\(=10-3i + 4\)

Step3: Combine like terms

\(=14-3i\)

g. \(i(-3 + 2i)\)

Step1: Distribute \(i\)

\(-3i+2i^{2}\)

Step2: Recall that \(i^{2}=-1\)

\(=-3i+2\times(-1)\)
\(=-2-3i\)

h. \(5i(2 - 4i)\)

Step1: Distribute \(5i\)

\(10i-20i^{2}\)

Step2: Recall that \(i^{2}=-1\)

\(=10i-20\times(-1)\)
\(=20 + 10i\)

i. \((4i)(2i)(3 - 4i)\)

Step1: Multiply \((4i)(2i)\) first

\(8i^{2}\)

Step2: Recall that \(i^{2}=-1\)

\(8\times(-1)=-8\)

Step3: Multiply \(-8\) with \((3 - 4i)\)

\(-8\times3-8\times(-4i)\)
\(=-24 + 32i\)

Answer:

s

Part 5

• First Graph: B. 2 Real Solutions & I. Discriminant is positive
• Second Graph: C. 2 Complex Solutions & II. Discriminant is negative
• Third Graph: A. 1 Real Solution & III. Discriminant is zero

Part 6

a. \(\boldsymbol{7 - 4i}\)

b. \(\boldsymbol{-1 + 8i}\)

c. \(\boldsymbol{7 - 3i}\)

d. \(\boldsymbol{-3 + 5i}\)

e. \(\boldsymbol{24 - 10i}\)

f. \(\boldsymbol{14 - 3i}\)

g. \(\boldsymbol{-2 - 3i}\)

h. \(\boldsymbol{20 + 10i}\)

i. \(\boldsymbol{-24 + 32i}\)