Let's fill out the table with a detailed step-by-step solution for converting each given equation into its standard form and identifying the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] for each quadratic equation.
### Equation 1: [tex]\( 4(x^2 + 2x - 4) = 0 \)[/tex]
1. Use the distributive property to multiply 4 with each term inside the parentheses:
[tex]\[
4 \cdot x^2 + 4 \cdot 2x + 4 \cdot (-4) = 0
\][/tex]
2. Simplify:
[tex]\[
4x^2 + 8x - 16 = 0
\][/tex]
3. Identify the coefficients:
[tex]\[
\begin{align*}
a &= 4 \\
b &= 8 \\
c &= -16
\end{align*}
\][/tex]
### Equation 2: [tex]\( 2x^2 + 5 = 3x \)[/tex]
1. Subtract [tex]\(3x\)[/tex] from both sides to bring all terms to one side, creating a standard form:
[tex]\[
2x^2 + 5 - 3x = 0
\][/tex]
2. Rearrange the terms:
[tex]\[
2x^2 - 3x + 5 = 0
\][/tex]
3. Identify the coefficients:
[tex]\[
\begin{align*}
a &= 2 \\
b &= -3 \\
c &= 5
\end{align*}
\][/tex]
### Equation 3: [tex]\( x^2 + 8x + 4 = 6 \)[/tex]
1. Subtract 6 from both sides to bring all terms to one side, creating a standard form:
[tex]\[
x^2 + 8x + 4 - 6 = 0
\][/tex]
2. Simplify:
[tex]\[
x^2 + 8x - 2 = 0
\][/tex]
3. Identify the coefficients:
[tex]\[
\begin{align*}
a &= 1 \\
b &= 8 \\
c &= -2
\end{align*}
\][/tex]
### Equation 4: [tex]\( 3(x^2 - 2x + 4) = 0 \)[/tex]
1. Use the distributive property to multiply 3 with each term inside the parentheses:
[tex]\[
3 \cdot x^2 + 3 \cdot (-2x) + 3 \cdot 4 = 0
\][/tex]
2. Simplify:
[tex]\[
3x^2 - 6x + 12 = 0
\][/tex]
3. Identify the coefficients:
[tex]\[
\begin{align*}
a &= 3 \\
b &= -6 \\
c &= 12
\end{align*}
\][/tex]
### Completed Table:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
Given Equation & Steps Needed & Standard Form & $a$, $b$, $c$ \\
\hline
\multirow{2}{}{$4(x^2 + 2x - 4) = 0$} & \multirow{2}{}{\begin{tabular}{l}
Use Distributive \\
property. Multiply 4 to \\
all terms enclosed.
\end{tabular}}
& \multirow{2}{*}{$4x^2 + 8x - 16 = 0$}
& $a = 4$ \\
& & & $b = 8$ \\
& & & $c = -16$ \\
\hline
\multirow{3}{*}{$2x^2 + 5 = 3x$}
& \multirow{3}{*}{\begin{tabular}{l}
Addition property \\
Add $-3x$ to both sides \\
of the equation.
\end{tabular}}
& \multirow{3}{*}{$2x^2 - 3x + 5 = 0$}
& $a = 2$ \\
& & & $b = -3$ \\
& & & $c = 5$ \\
\hline
\multirow{3}{*}{$x^2 + 8x + 4 = 6$}
& \multirow{3}{*}{\begin{tabular}{l}
Subtract 6 from \\
both sides.
\end{tabular}}
& \multirow{3}{*}{$x^2 + 8x - 2 = 0$}
& $a = 1$ \\
& & & $b = 8$ \\
& & & $c = -2$ \\
\hline
\multirow{2}{*}{$3(x^2 - 2x + 4) = 0$}
& \multirow{2}{*}{\begin{tabular}{l}
Use Distributive \\
property. Multiply 3 to \\
all terms enclosed.
\end{tabular}}
& \multirow{2}{*}{$3x^2 - 6x + 12 = 0$}
& $a = 3$ \\
& & & $b = -6$ \\
& & & $c = 12$ \\
\hline
\end{tabular}
\][/tex]
