Question 189673
X-Intercept(s):



Remember, the x-intercept(s) occur when f(x) (or y) is equal to zero.




{{{f(x)= x^2-4x-32}}} Start with the given function.



{{{0= x^2-4x-32}}} Plug in {{{f(x)=0}}}



Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=1}}}, {{{b=-4}}}, and {{{c=-32}}}



Let's use the quadratic formula to solve for x



{{{x = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{x = (-(-4) +- sqrt( (-4)^2-4(1)(-32) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=-4}}}, and {{{c=-32}}}



{{{x = (4 +- sqrt( (-4)^2-4(1)(-32) ))/(2(1))}}} Negate {{{-4}}} to get {{{4}}}. 



{{{x = (4 +- sqrt( 16-4(1)(-32) ))/(2(1))}}} Square {{{-4}}} to get {{{16}}}. 



{{{x = (4 +- sqrt( 16--128 ))/(2(1))}}} Multiply {{{4(1)(-32)}}} to get {{{-128}}}



{{{x = (4 +- sqrt( 16+128 ))/(2(1))}}} Rewrite {{{sqrt(16--128)}}} as {{{sqrt(16+128)}}}



{{{x = (4 +- sqrt( 144 ))/(2(1))}}} Add {{{16}}} to {{{128}}} to get {{{144}}}



{{{x = (4 +- sqrt( 144 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (4 +- 12)/(2)}}} Take the square root of {{{144}}} to get {{{12}}}. 



{{{x = (4 + 12)/(2)}}} or {{{x = (4 - 12)/(2)}}} Break up the expression. 



{{{x = (16)/(2)}}} or {{{x =  (-8)/(2)}}} Combine like terms. 



{{{x = 8}}} or {{{x = -4}}} Simplify. 



So the answers are {{{x = 8}}} or {{{x = -4}}} 

  

This means that the x-intercepts are (8,0) and (-4,0)