Solve the following? #(x−1)(x−3)(x−5)(x+1)=−12#

2 Answers
Dan Kemp
Apr 22, 2018

#x=−0.645751,0.267949,3.732051,4.645751#

Explanation:

#(x-1)(x-3)(x-5)(x+1)=-12#

first, we can expand each of the brackets using FOIL

#(x-1)(x-3) = x^2 - 3x - x + 3#

#(x-5)(x+1) = x^2 +x -5x - 5#

Now we can simplify these and combine them back together into the original expression.

#(x^2 - 4x + 3) xx (x^2 -4x - 5) = -12#

Now we can factorise this.

#(color(lime)(x^2 - 4x + 3)) (color(skyblue)(x^2 -4x - 5)) = (color(lime)(x^2)) (color(skyblue)(x^2)) + (color(lime)(x^2))(color(skyblue)(-4x)) + (color(lime)(x^2))(color(skyblue)(-5)) + (color(lime)(-4x))(color(skyblue)(x^2)) + (color(lime)(-4x))(color(skyblue)(-4x)) + (color(lime)(-4x))(color(skyblue)(-5)) + (color(lime)(3))(color(skyblue)(x^2)) + (color(lime)(3))(color(skyblue)(-4x)) + (color(lime)(3))(color(skyblue)(-5))#

# = (x^4) + (-4x^3) + (-5x^2) + (-4x^3) + (16x^2) + (20x) + (3x^2) + (-12x) + (-15)#

Now we can collect like terms and simplify.

# = x^4 -4x^3 - 4x^3 -5x^2 + 3x^2 + 16x^2 + 20x -12x -15#

#x^4 - 8x^3 + 14x^2 + 8x -15 = -12#

Now we can find #x#

#x^4 - 8x^3 + 14x^2 + 8x color(red)(cancel(color(black)(-15) +15)) = -12 color(red)(+15)#

#x^4 - 8x^3 + 14x^2 + 8x = 3#

The only way that I could find to solve this is through using the Quartic formula.

#color(blue)(x=−0.645751,0.267949,3.732051,4.645751)#

Parabola
Apr 27, 2018

#+-sqrt3+2# and #+-sqrt7+2#

Explanation:

We have:

#(x-1)(x-3)(x-5)(x+1)=-12#

Multiply the first two and the last two parentheses together.

#=>(x^2-4x+3)(x^2-4x-5)=-12#

Hmm... We see that the first quadratic expression is 8 larger than the second.

We let #s=(x^2-4x+3)#

We now have:

#s(s-8)=-12#

#=>s^2-8s=-12# complete the square

#=>(s^2-8s+16)-16=-12#

#=>(s-4)^2=4#

#=>-2=s-4=2#

#=>2=s=6#

We can use this to solve for #x#.

When #s=2#...

#x^2-4x+3=2#

#=>x^2-4x+1=0#

#=>(x^2-4x+4)-4+1=0#

#=>(x-2)^2-3=0#

#=>(x-2)^2=3#

#=>x-2=+-sqrt(3)#

#=>x=+-sqrt(3)+2#

When #s=6#...

#x^2-4x+3=6#

#=>x^2-4x-3=0#

#=>(x^2-4x+4)-4-3=0#

#=>(x-2)^2-7=0#

#=>(x-2)^2=7#

#=>x-2=+-sqrt(7)#

#=>x=+-sqrt(7)+2#

The answers are #+-sqrt3+2# and #+-sqrt7+2#

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