Test signed algebra and application choices

1. 1
   Remove the constant

   Subtract 4 from both sides.

   $$-0.5x=3.5$$
2. 2
   Divide by the signed coefficient

   A positive divided by a negative is negative.

   $$x=-7$$

Independent check: Negative 0.5 times negative 7 plus 4 equals 7.5.

1. 1
   Expand and combine

   Distribute 7 and 4, then simplify the left constants.

   $$7x-11=4x+4$$
2. 2
   Collect and solve

   Subtract 4x and add 11.

   $$3x=15\Rightarrow x=5$$

Independent check: Each original side equals 24 when x is 5.

1. 1
   Identify the GCF

   Fourteen and 21 share 7, and both terms contain x cubed.

   $$\gcd(14x^4,21x^3)=7x^3$$
2. 2
   Factor

   Divide each term by the GCF.

   $$14x^4+21x^3=7x^3(2x+3)$$

Independent check: Distributing 7x cubed returns the original polynomial.

1. 1
   Recognize the cubes

   Eight x cubed is (2x) cubed.

   $$8x^3-1=(2x)^3-1^3$$
2. 2
   Apply the pattern

   Use (a - b)(a squared + ab + b squared).

   $$(2x-1)(4x^2+2x+1)$$

Independent check: The middle cubic terms cancel when the factors are multiplied.

1. 1
   Factor

   Use positive 5 and negative 3.

   $$(x+5)(x-3)=0$$
2. 2
   Solve each factor

   Apply the zero-product property.

   $$x=-5\quad\text{or}\quad x=3$$

Independent check: Both values make the polynomial zero, and their sum is negative 2 as required by negative b/a.

1. 1
   Compute the discriminant

   Use a = 2, b = negative 4, and c = 5.

   $$\Delta=(-4)^2-4(2)(5)=-24$$
2. 2
   Apply and simplify

   Square root negative 24 is 2i square root 6.

   $$x=\frac{4\pm2i\sqrt6}{4}=1\pm\frac{\sqrt6}{2}i$$

Independent check: The conjugate roots have sum 2 and product 5/2, matching the coefficient ratios.

1. 1
   Multiply all term pairs

   Compute the first, outer, inner, and last products.

   $$2x^2+8x-3x-12$$
2. 2
   Combine the middle terms

   Eight x minus three x is five x.

   $$2x^2+5x-12$$

Independent check: At x = 1, both forms equal negative 5.

1. 1
   Record the restriction

   The denominator cannot be zero.

   $$x\ne2$$
2. 2
   Clear and solve

   Multiply by x minus 2 and collect x.

   $$x+1=2x-4\Rightarrow x=5$$

Independent check: The original fraction becomes 6/3, which equals 2.

1. 1
   Identify a and b

   The leading coefficient is negative 2 and the linear coefficient is 8.

   $$a=-2,\qquad b=8$$
2. 2
   Use the axis formula

   Substitute into negative b over 2a.

   $$x=-\frac{8}{2(-2)}=2$$

Independent check: The values f(1) and f(3) are equal, confirming symmetry around x = 2.

1. 1
   Write standard form

   Expand and subtract 54.

   $$x^2+3x-54=0$$
2. 2
   Factor and interpret

   The roots are 6 and negative 9, and only 6 is a valid width.

   $$(x+9)(x-6)=0\Rightarrow x=6,\quad x+3=9$$

Independent check: The dimensions differ by 3 and their product is 54.