The first two steps in determining the solution set of the system of equations [tex]y = x^2 - 6x + 12[/tex] and [tex]y = 2x - 4[/tex] algebraically are shown in the table.

\[
\begin{tabular}{|c|c|}
\hline
Step & Equation \\
\hline
Step 1 & [tex]x^2 - 6x + 12 = 2x - 4[/tex] \\
\hline
Step 2 & [tex]x^2 - 8x + 16 = 0[/tex] \\
\hline
\end{tabular}
\]

Which represents the solution(s) of this system of equations?

A. [tex](4, 4)[/tex]
B. [tex](-4, -12)[/tex]
C. [tex](4, 4)[/tex] and [tex](-4, 12)[/tex]
D. [tex](-4, 4)[/tex] and [tex](4, 12)[/tex]



Answer :

To determine the solution set of the system of equations [tex]\( y = x^2 - 6x + 12 \)[/tex] and [tex]\( y = 2x - 4 \)[/tex] algebraically, follow these steps:

### Step 1: Set the equations equal to each other
Given the equations [tex]\( y = x^2 - 6x + 12 \)[/tex] and [tex]\( y = 2x - 4 \)[/tex], set them equal to each other to find the values of [tex]\( x \)[/tex] where the two equations intersect:
[tex]\[ x^2 - 6x + 12 = 2x - 4 \][/tex]

### Step 2: Rearrange into a standard form quadratic equation
To create a standard form quadratic equation (i.e., [tex]\( ax^2 + bx + c = 0 \)[/tex]), move all terms to one side of the equation:
[tex]\[ x^2 - 6x + 12 - 2x + 4 = 0 \][/tex]
Combine like terms:
[tex]\[ x^2 - 8x + 16 = 0 \][/tex]

### Step 3: Solve the quadratic equation
Solve the quadratic equation [tex]\( x^2 - 8x + 16 = 0 \)[/tex].

### Roots of the quadratic equation
The roots of the equation [tex]\( x^2 - 8x + 16 = 0 \)[/tex] will give us the x-values where the solutions occur. In this case, the roots are:
[tex]\[ x = 4 \][/tex]
(since)[tex]$(x-4)^2=0$[/tex]

### Step 4: Find the corresponding y-values
Substitute [tex]\( x = 4 \)[/tex] back into either of the original equations to find the corresponding y-values.
Using the equation [tex]\( y = 2x - 4 \)[/tex]:
[tex]\[ y = 2(4) - 4 = 8 - 4 = 4 \][/tex]

This means the solution set for the system of equations [tex]\( y = x^2 - 6x + 12 \)[/tex] and [tex]\( y = 2x - 4 \)[/tex] is:
[tex]\[ (4, 4) \][/tex]

Therefore, the correct solution(s) of this system of equations are:
[tex]\[ (4, 4) \][/tex]

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