ACT Problem: Finding The x-Intercept Of Given Line

[816318]

What is the x-intercept of the graph of y = x2 – 4x + 4?

How would you foil this?

 

[mathlearn]

Re: ACT problem

To find the x intercept of the graph of the function $y = x^2 – 4x + 4$

We may either use the completing the square method or we may use the formula $x=\frac{-b}{2a}$

The easiest way is to use $x=\frac{-b}{2a}$

From $y = x^2 – 4x + 4$ which is in the form of $ax^2+bx+c$ we can find the values for $b$ and $a$ as $b=-4$ & $a=1$

$-b$ in the formula stands for the opposite of the value of $b$ in the above form.

$x=\frac{-(-4)}{2*1}$
$x=\frac{4}{2}$
$x=2$

using complete the square method to form the graph of the function of the form $y=\pm(x+b)^2+c$ or the vertex form

$y= (x^2 – 4x+(\frac{b}{2})^2 ) + 4 - (\frac{b}{2})^2 ) $
$ y=(x^2 – 4x+(\frac{-4}{2})^2 ) + 4 - (\frac{-4}{2})^2 ) $
$ y=(x^2 – 4x+4 ) + 4 - 4 $
$y=(x-2)^2$

which now the x means -b which is 2.

Now it can be seen using Desmos one $x$ intercept of both the forms is $(2,0)$

[graph]gf10si3evs[/graph]

 

[MarkFL]

Re: ACT problem

What is the x-intercept of the graph of y = x2 – 4x + 4?

How would you foil this?

To find the $x$-intercept, we can set $y=0$ and solve for $x$:

\(\displaystyle x^2-4x+4=0\)

To factor, we need to look for two factors of 4 whose sum is -4, and we find:

\(\displaystyle (-2)(-2)=4\)

\(\displaystyle (-2)+(-2)=-4\)

Thus, the factored form is:

\(\displaystyle (x-2)(x-2)=0\)

or:

\(\displaystyle (x-2)^2=0\)

We have a repeated root, of multiplicity 2. Since the multiplicity is even, we know the graph will touch the $x$-axis without passing through it. Equating this factor to zero, we find:

\(\displaystyle x-2=0\)

\(\displaystyle x=2\)

Thus, we know the given graph has one $x$-intercept at $(2,0)$.

 

[Greg]

Re: ACT problem

$$x^2-4x+4=x^2-2x-2x+4=x(x-2)-2(x-2)=(x-2)(x-2)=0\implies x=2$$