Let's rewrite each given quadratic equation in standard form, [tex]\( ax^2 + bx + c = 0 \)[/tex], and identify the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] for each equation:
### 1. [tex]\( x^2 + 3x = 7 \)[/tex]
Rewriting this equation in standard form:
[tex]\[ x^2 + 3x - 7 = 0 \][/tex]
So, the coefficients are:
[tex]\[ a = 1, \quad b = 3, \quad c = -7 \][/tex]
### 2. [tex]\( -2x^2 = 4x \)[/tex]
Rewriting this equation in standard form:
[tex]\[ -2x^2 - 4x = 0 \][/tex]
So, the coefficients are:
[tex]\[ a = -2, \quad b = -4, \quad c = 0 \][/tex]
### 3. [tex]\( 4x^2 = 121 \)[/tex]
Rewriting this equation in standard form:
[tex]\[ 4x^2 - 121 = 0 \][/tex]
So, the coefficients are:
[tex]\[ a = 4, \quad b = 0, \quad c = -121 \][/tex]
### 4. [tex]\( 3 + 5x = -9x^2 \)[/tex]
Rewriting this equation in standard form:
[tex]\[ 9x^2 + 5x + 3 = 0 \][/tex]
So, the coefficients are:
[tex]\[ a = 9, \quad b = 5, \quad c = 3 \][/tex]
### 5. [tex]\( 3x^2 - 14x = x^2 + 12 \)[/tex]
Rewriting this equation in standard form:
[tex]\[ 3x^2 - 14x - (x^2 + 12) = 0 \][/tex]
[tex]\[ 2x^2 - 14x - 12 = 0 \][/tex]
So, the coefficients are:
[tex]\[ a = 2, \quad b = -14, \quad c = -12 \][/tex]
Let's compile this into the table format you provided:
\begin{tabular}{|l|l|r|r|r|}
\hline
Quadratic Equation & Standard form & [tex]\(a\)[/tex] & [tex]\(b\)[/tex] & [tex]\(c\)[/tex] \\
\hline
1. [tex]\(x^2 + 3x = 7\)[/tex] & [tex]\(x^2 + 3x - 7 = 0\)[/tex] & 1 & 3 & -7 \\
\hline
2. [tex]\(-2x^2 = 4x\)[/tex] & [tex]\(-2x^2 - 4x = 0\)[/tex] & -2 & -4 & 0 \\
\hline
3. [tex]\(4x^2 = 121\)[/tex] & [tex]\(4x^2 - 121 = 0\)[/tex] & 4 & 0 & -121 \\
\hline
4. [tex]\(3 + 5x = -9x^2\)[/tex] & [tex]\(9x^2 + 5x + 3 = 0\)[/tex] & 9 & 5 & 3 \\
\hline
5. [tex]\(3x^2 - 14x = x^2 + 12\)[/tex] & [tex]\(2x^2 - 14x - 12 = 0\)[/tex] & 2 & -14 & -12 \\
\hline
\end{tabular}
