How do you solve #(4x-2)/(x-6) = -x/(x+5)# and find any extraneous solutions?

Question

Avery Alexander · Accepted Answer

The solns. are #x_1~=0.655, and, x_2~=-3.055#.

#(4x-2)/(x-6)=-x/(x+5)# By Cross-Multiplication, #(4x-2)(x+5)=-x(x-6)# By Expansion, # 4x^2+20x-2x-10=-x^2+6x# By Transposition, #4x^2+18x-10+x^2-6x=0# #:. 5x^2+12x-10=0.............(1)# We use the Formula to find the roots of this Qudr. Eqn. :- The roots #alpha, beta# of the Gen. Qudr. Eqn. #ax^2+bx+c=0# are #alpha=(-b+sqrt(b^2-4ac))/(2a), beta=(-b-sqrt(b^2-4ac))/(2a)# In our case, since, #a=5, b=12, c=-10#, we get, #alpha=(-12+sqrt(12^2-4*5*(-10)))/(2*5)=(-12+sqrt(144+200))/10# #:. alpha=(-12+sqrt344)/10, and, beta=(-12-sqrt344)/10# Taking, #sqrt344~=18.55, alpha~=0.655, and, beta~=-3.055# Thus, the solns. are #x_1=alpha~=0.655, and, x_2=beta~=-3.055#.

Ethan Adkins · Answer

undefined To solve the equation (4x-2)/(x-6) = -x/(x+5) and find any extraneous solutions:

1. Cross multiply to eliminate the fractions: (4x - 2)(x + 5) = -x(x - 6).
2. Expand and simplify both sides of the equation: 4x^2 + 20x - 2x - 10 = -x^2 + 6x.
3. Combine like terms: 4x^2 + 18x - 10 = -x^2 + 6x.
4. Move all terms to one side to set the equation to zero: 4x^2 + 18x - 10 + x^2 - 6x = 0.
5. Combine like terms again: 5x^2 + 12x - 10 = 0.
6. Solve the quadratic equation by factoring, completing the square, or using the quadratic formula.
7. After obtaining solutions, check each solution in the original equation to ensure they are not extraneous. If any solution makes the denominator zero, it is extraneous.