Question

(x+1)(x+2)-(x-3)(x+4)=6(3x-1)(2x+7)-(x+1)(6x-5)=7​

Answer 1

$(x+1)(x+2)-(x-3)(x+4)=6\\x^2+2x+x+2-(x^2+4x-3x-12)=6\\x^2+2x+x+2-(x^2+x-12)=6\\x^2+2x+x+2-x^2-x+12=6\\2x+14=6\\2x=6-14\\2x=-8\\x=-4$

$(3x-1)(2x+7)-(x+1)(6x-5)=7\\6x^2+21x-2x-7-(6x^2-5x+6x-5)=7\\6x^2+21x-2x-7-(6x^2+x-5)=7\\6x^2+21x-2x-7-6x^2-x+5=7\\18x-2=7\\18x=7+2\\18x=9\\x=\frac{1}{2}$

Answer 2

To solve the equation, let's break it down step by step.

Step 1: Expand the expressions First, expand the expressions on both sides of the equation:

(x+1)(x+2) - (x-3)(x+4) = 6(3x-1)(2x+7) - (x+1)(6x-5) = 7

Expanding the left-hand side (LHS): (x+1)(x+2) = x(x+2) + 1(x+2) = x^2 + 2x + x + 2 = x^2 + 3x + 2

-(x-3)(x+4) = -x(x+4) + 3(x+4) = -x^2 - 4x + 3x + 12 = -x^2 - x + 12

So, the LHS is: LHS = x^2 + 3x + 2 - (x^2 + x - 12) = x^2 + 3x + 2 - x^2 - x + 12 = 2x + 14

Expanding the right-hand side (RHS): 6(3x-1)(2x+7) = 6(6x^2 + 17x - 7) = 36x^2 + 102x - 42

-(x+1)(6x-5) = -(6x^2 - 5x + 6x - 5) = -6x^2 + 11x + 5

So, the RHS is: RHS = 36x^2 + 102x - 42 - (-6x^2 + 11x + 5) = 36x^2 + 102x - 42 + 6x^2 - 11x - 5 = 42x^2 + 91x - 47

Step 2: Set the expressions equal to each other Now that we have the LHS and RHS expressions, set them equal to each other:

2x + 14 = 42x^2 + 91x - 47

Step 3: Simplify and arrange the equation in standard form Move all terms to one side of the equation:

42x^2 + 91x - 2x - 14 - 47 = 0

Simplify:

42x^2 + 89x - 61 = 0

Now we have a quadratic equation in standard form: ax^2 + bx + c = 0 where a = 42, b = 89, and c = -61.

Step 4: Solve the quadratic equation To solve the quadratic equation, you can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Plugging in the values, we get:

x = (-(89) ± √((89)^2 - 4(42)(-61))) / 2(42)

x = (-89 ± √(7921 + 10164)) / 84

x = (-89 ± √(18085)) / 84

x = (-89 ± 134.48) / 84

Now, we have two possible solutions:

1. x = (-89 + 134.48) / 84 = 45.48 / 84 ≈ 0.5414
2. x = (-89 - 134.48) / 84 = -223.48 / 84 ≈ -2.6607

So, the two solutions for the equation are approximately x ≈ 0.5414 and x ≈ -2.6607.