Respuesta :
Answer:
The solutions [tex]x^4-5x^2-36=0[/tex] are [tex]x=3,\:x=-3,\:x=2i,\:x=-2i[/tex] and the x-intercepts of [tex]y=x^4-5x^2-36[/tex] are [tex]x=3,\:x=-3[/tex]
Step-by-step explanation:
Finding the solutions to [tex]x^4-5x^2-36[/tex] means finding the roots, a root is where the function is equal to zero.
The x-intercept is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero.
To find the roots you need to:
Rewrite the equation with [tex]u=x^2[/tex] and [tex]u^2=x^4[/tex]
[tex]u^2-5u-36=0[/tex]
Solve by factoring
[tex]\mathrm{Break\:the\:expression\:into\:groups}\\u^2-5u-36=\left(u^2+4u\right)+\left(-9u-36\right)[/tex]
[tex]\mathrm{Factor\:out\:}u\mathrm{\:from\:}u^2+4u=u\left(u+4\right)[/tex]
[tex]\mathrm{Factor\:out\:}-9\mathrm{\:from\:}-9u-36=-9\left(u+4\right)[/tex]
[tex]u^2-5u-36=u\left(u+4\right)-9\left(u+4\right)[/tex]
[tex]\mathrm{Factor\:out\:common\:term\:}u+4\\\left(u+4\right)\left(u-9\right)[/tex]
[tex]u^2-5u-36=\left(u+4\right)\left(u-9\right)=0[/tex]
Using the Zero factor Theorem: if ab = 0 then a = 0 or b = 0 (or both a = 0 and b = 0)
The solutions to the quadratic equation are:
[tex]\:u=-4,\:u=9[/tex]
Substitute back [tex]u=x^2[/tex], solve for x
[tex]x^4-5x^2-36=(u-9)(u+4)=(x^2-9)(x^2+4)[/tex]
Apply the difference of squares formula
[tex]x^4-5x^2-36=(x^2-9)(x^2+4)=(x-3)(x+3)(x^2+4)[/tex]
[tex](x-3)(x+3)(x^2+4)=0[/tex]
Using the Zero factor Theorem: if ab = 0 then a = 0 or b = 0 (or both a = 0 and b = 0)
The solutions are:
[tex]\:x=3,\:x=-3,\:x=2i,\:x=-2i[/tex]
Because two of the solutions are complex roots the only x-intercepts are [tex]x=3,\:x=-3[/tex]
