Answer:
There are two points common with the x-axis
The vertex is under the x-axis by 24 units at x = 6
Step-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
