Without drawing the graph of the equation answer the question. How many points does the given equation have in common with the x-axis and where is the vertex in relation to the x-axis? y=x^2-12x+12

Answer:There are two points common with the x-axis The vertex is under the x-axis by 24 units at x = 6Step-by-step explanation:* For the quadratic equation y = ax² + bx + c- The roots of the equation are the intersection point between   the equation and the x-axis ⇒ y = 0- To find these roots we factorize the equation into two factors   and equate each factor with zero - The graph of the quadratic equation is called parabola,  the parabola has vertex point. If the vertex point is (h , k)∴ h = -b/2a, where b is the coefficient of x and is the coefficient of x²∴ k = y where x = h* Lets solve the problem∵ y = x² - 12x + 12- The formula to find the values of x when y = 0 is  [tex]x=\frac{-b+\sqrt{b^{2}-4ac}}{2a},x=\frac{-b-\sqrt{b^{2}-4ac}}{2a}[/tex]∵ a = 1 , b = -12 , c = 12∴ [tex]x=\frac{-(-12)+\sqrt{(-12)^{2}-4(1)(12)}}{2(1)}=\frac{12+\sqrt{144-48}}{2}[/tex]∴ [tex]x=\frac{12+\sqrt{96}}{2}=\frac{12+4\sqrt{6}}{2}=6+2\sqrt{6}[/tex]∴ The two roots are 6 + 2√6 and 6 - 2√6* There are two points common with the x-axis* Lets calculate the vertex∵ h = -b/2a∵ b = -12 and a = 1∴ h =-(-12)/2(1) = 12/2 = 6∴ k = (6²) - 12(6) + 12 = 36 - 72 + 12 = -24∴ The vertex is (6 , -24)* That means the vertex is under the x-axis by 24 units at x = 6