Question

1. rewrite the expression: (3 + 6i)(7 - 2i)

1 point (c.cn.a.2)

1. solve the quadratic equation:

(3x - 2)² = -4(2x - 1)
1 point (a.rei.b.4)

Explanation:

Problem 4: Rewrite the expression \((3 + 6i)(7 - 2i)\)

Step 1: Use the distributive property (FOIL method)

Multiply each term in the first complex number by each term in the second complex number:
\[

$$\begin{align*} (3 + 6i)(7 - 2i)&=3\times7+3\times(-2i)+6i\times7+6i\times(-2i)\\ &=21 - 6i + 42i - 12i^2 \end{align*}$$

\]

Step 2: Simplify using \(i^2=-1\)

Recall that \(i^2 = - 1\), so substitute \(-1\) for \(i^2\) in the expression:
\[

$$\begin{align*} 21 - 6i + 42i - 12i^2&=21 - 6i + 42i - 12\times(-1)\\ &=21 - 6i + 42i + 12 \end{align*}$$

\]

Step 3: Combine like terms

Combine the real parts and the imaginary parts separately:
\[

$$\begin{align*} 21+12+(-6i + 42i)&=(21 + 12)+( - 6 + 42)i\\ &=33 + 36i \end{align*}$$

\]

Step 1: Expand both sides of the equation

First, expand \((3x - 2)^2\) using the formula \((a - b)^2=a^2-2ab + b^2\) where \(a = 3x\) and \(b = 2\), and expand \(-4(2x - 1)\) using the distributive property:
\[

$$\begin{align*} (3x)^2-2\times3x\times2+2^2&=-4\times2x+(-4)\times(-1)\\ 9x^2-12x + 4&=-8x + 4 \end{align*}$$

\]

Step 2: Move all terms to one side to form a standard quadratic equation

Subtract \(-8x + 4\) from both sides to get:
\[

$$\begin{align*} 9x^2-12x + 4+8x - 4&=0\\ 9x^2-4x&=0 \end{align*}$$

\]

Step 3: Factor out the greatest common factor (GCF)

The GCF of \(9x^2\) and \(-4x\) is \(x\), so factor out \(x\):
\[
x(9x - 4)=0
\]

Step 4: Use the zero - product property

If \(ab = 0\), then either \(a = 0\) or \(b = 0\). So we set each factor equal to zero and solve for \(x\):

• If \(x = 0\), then the equation is satisfied.
• If \(9x-4 = 0\), then \(9x=4\), and \(x=\frac{4}{9}\)

Answer:

\(33 + 36i\)

Problem 7: Solve the quadratic equation \((3x - 2)^2=-4(2x - 1)\)