Question

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 1

There are two points common with the x-axis

The vertex is under the x-axis by 24 units at x = 6

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

`x=\frac{-b+\sqrt{b^(2)-4ac}}{2a},x=\frac{-b-\sqrt{b^(2)-4ac}}{2a}`

∵ a = 1 , b = -12 , c = 12

∴
`x=\frac{-(-12)+\sqrt{(-12)^(2)-4(1)(12)}}{2(1)}=(12+√(144-48))/(2)`

∴
`x=(12+√(96))/(2)=(12+4√(6))/(2)=6+2√(6)`

∴ 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