Question

How do you find the vertex and the intercepts for \[9{x^2} - 12x + 4 = 0\]?

Answer 1

Hint:
We can find the vertex and intercepts of the equation by their general formulas obtained and to find the y-intercept of the given equation, just we need to substitute \[x\]= 0 in the given equation and solve for y and to find the x-intercept of the given equation, just we need to substitute y = 0 in the given equation and solve for x.

Complete step by step solution:
The given equation is
\[9{x^2} - 12x + 4 = 0\]
The given equation is in the form of \[a{x^2} + bx + c\], in which we need to find the vertex of x and y coordinate.
The equation of parabola in vertex form is
\[y = a{\left( {x - h} \right)^2} + k\]
In which h and k are the coordinates of the vertex and a is the multiplier.
Let us solve this quadratic equation by completing the square,
Divide both sides of the equation by 9 to have 1 as the coefficient of the first term:
\[\dfrac{{{x^2}}}{9} - \dfrac{{12}}{9}x + \dfrac{4}{9} = 0\]
\[{x^2} - \dfrac{4}{3}x + \dfrac{4}{9} = 0\]
The coefficient of the \[{x^2}\]term must be 1, hence factor out of 9 we get
\[9\left( {{x^2} - \dfrac{4}{3}x + \dfrac{4}{9}} \right) = 0\] ………….. 1
 Subtract \[\dfrac{4}{9}\]to both side of the equation we get:
\[{x^2} - \dfrac{4}{3}x + \dfrac{4}{9} - \dfrac{4}{9} = - \dfrac{4}{9}\]
\[{x^2} - \dfrac{4}{3}x = - \dfrac{4}{9}\]
Take the coefficient of x, which is \[\dfrac{4}{3}\], divide by two, giving \[\dfrac{2}{3}\], and finally square it giving \[ - \dfrac{4}{9}\].
From equation 1 we get
\[9\left( {{x^2} + 2\left( { - \dfrac{2}{3}} \right)x + \dfrac{9}{4} - \dfrac{9}{4} + \dfrac{9}{4}} \right) = 0\]
Implies that,
\[9{\left( {x - \dfrac{2}{3}} \right)^2} + 0 = 0\]
\[9{\left( {x - \dfrac{2}{3}} \right)^2} = 0\]
As the equation we got is in vertex from as
\[y = a{\left( {x - h} \right)^2} + k\]
In which h = \[\dfrac{2}{3}\] and k = 0.
Therefore, the vertex we got is \[\left( {\dfrac{2}{3},0} \right)\]
Let us find the intercepts for the obtained equation
\[9{\left( {x - \dfrac{2}{3}} \right)^2} = 0\]
Solving for x-intercept, hence we get
\[x = \dfrac{2}{3}\]

Therefore, x-intercept is at \[x = \dfrac{2}{3}\].

Note:
As per the given equation consists of x and y terms based on the intercept asked, we need to solve for it. For ex if y-intercept is asked substitute x=0 and solve for y and if x-intercept is asked substitute y=0 and solve for x and the y-intercept of an equation is a point where the graph of the equation intersects the y-axis.