Question

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

Answer 1

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.