The equation you are asked to solve is:

$x^2 - 8x + 12 = 0$

Steps to Solve by Graphing:

1. Find the vertex of the parabola. This equation is in the form $ax^2 + bx + c$, where $a = 1$, $b = -8$, and $c = 12$.

   • The x-coordinate of the vertex is found using the formula: $x = \frac{-b}{2a} = \frac{-(-8)}{2(1)} = \frac{8}{2} = 4$
   • To find the y-coordinate of the vertex, substitute $x = 4$ back into the original equation: $y = (4)^2 - 8(4) + 12 = 16 - 32 + 12 = -4$ So, the vertex is at $(4, -4)$.

2. Plot additional points: Pick values of $x$ on either side of $x = 4$ to find corresponding $y$-values:

   • For $x = 3$: $y = (3)^2 - 8(3) + 12 = 9 - 24 + 12 = -3$ So, point $(3, -3)$.
   • For $x = 5$: $y = (5)^2 - 8(5) + 12 = 25 - 40 + 12 = -3$ So, point $(5, -3)$.
   • For $x = 2$: $y = (2)^2 - 8(2) + 12 = 4 - 16 + 12 = 0$ So, point $(2, 0)$.
   • For $x = 6$: $y = (6)^2 - 8(6) + 12 = 36 - 48 + 12 = 0$ So, point $(6, 0)$.

3. Graph the parabola: You should plot the points $(4, -4)$, $(3, -3)$, $(5, -3)$, $(2, 0)$, and $(6, 0)$ on the graph. The points $(2, 0)$ and $(6, 0)$ are the x-intercepts, which are the solutions to the equation.

Solution:

The solutions are $x = 2$ and $x = 6$.

Would you like any more details or clarifications?

Related Questions:

1. How can we derive the vertex form of a quadratic equation?
2. What are other methods for solving quadratic equations besides graphing?
3. How does the discriminant help in identifying the nature of solutions?
4. How does completing the square compare with solving quadratics by graphing?
5. What is the significance of the axis of symmetry in a parabola?

Tip:

When solving quadratics by graphing, always check the vertex and x-intercepts carefully to ensure accurate solutions.