To solve the system of equations given by [tex]\( y = x^2 - 6x + 12 \)[/tex] and [tex]\( y = 2x - 4 \)[/tex], we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously.
### Step-by-Step Solution:
1. Set the equations equal to each other:
Since both expressions are equal to [tex]\( y \)[/tex], we can set them equal to each other:
[tex]\[
x^2 - 6x + 12 = 2x - 4
\][/tex]
2. Rearrange to form a quadratic equation:
Move all terms to one side to simplify the equation:
[tex]\[
x^2 - 6x + 12 - 2x + 4 = 0
\][/tex]
Combining like terms, we get:
[tex]\[
x^2 - 8x + 16 = 0
\][/tex]
3. Solve the quadratic equation:
The equation [tex]\( x^2 - 8x + 16 = 0 \)[/tex] can be solved by factoring or using the quadratic formula. This specific equation is a perfect square trinomial, which can be factored as:
[tex]\[
(x - 4)^2 = 0
\][/tex]
Taking the square root of both sides, we have:
[tex]\[
x - 4 = 0 \implies x = 4
\][/tex]
4. Find the corresponding [tex]\( y \)[/tex] value:
Substitute [tex]\( x = 4 \)[/tex] into one of the original equations to find the corresponding [tex]\( y \)[/tex] value. Using [tex]\( y = 2x - 4 \)[/tex]:
[tex]\[
y = 2(4) - 4 = 8 - 4 = 4
\][/tex]
Thus, the point of intersection, and hence the solution to the system of equations, is:
[tex]\[
(x, y) = (4, 4)
\][/tex]
### Conclusion:
The solution to 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 answer is:
[tex]\[
(4, 4)
\][/tex]
