Test algebra structure before increasing difficulty

1. 1
   Collect like terms

   Subtract 2x and 9 from both sides.

   $$2x=-16$$
2. 2
   Divide

   Divide both sides by 2.

   $$x=-8$$

Independent check: Both sides equal negative 23 when x is negative 8.

1. 1
   Expand both sides

   Distribute the outer factors.

   $$6x-6=6x-6$$
2. 2
   Classify

   The resulting identity is true for every real x.

   $$0=0\Rightarrow x\in\mathbb R$$

Independent check: The two original sides are the same product written with different grouping.

1. 1
   Identify the common factor

   Both terms contain 5y.

   $$\gcd(15y^2,5y)=5y$$
2. 2
   Factor

   Divide each term by 5y.

   $$15y^2-5y=5y(3y-1)$$

Independent check: Distribution restores 15y squared minus 5y.

1. 1
   Choose the pair

   Use positive 4 and negative 3.

   $$4(-3)=-12,\qquad4+(-3)=1$$
2. 2
   Write the factors

   Place the two constants beside x.

   $$(x+4)(x-3)$$

Independent check: The outer and inner products combine to x when expanded.

1. 1
   Factor

   Use negative 2 and negative 5.

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

   Apply the zero-product property.

   $$x=2\quad\text{or}\quad x=5$$

Independent check: The roots have sum 7 and product 10, and both substitute to zero.

1. 1
   Find the discriminant

   Use b squared minus 4ac.

   $$\Delta=4^2-4(4)(5)=-64$$
2. 2
   Apply the formula

   Simplify the imaginary radical and divide by 8.

   $$x=\frac{-4\pm8i}{8}=-\frac12\pm i$$

Independent check: The conjugate roots have sum negative 1 and product 5/4, matching b/a and c/a.

1. 1
   Distribute

   Expand both products, keeping the negative sign.

   $$6x-15-2x-8$$
2. 2
   Combine like terms

   Combine the x-terms and constants separately.

   $$4x-23$$

Independent check: At x = 2, both the original and simplified expressions equal negative 15.

1. 1
   Record the restriction

   The denominator cannot equal zero.

   $$x\ne3$$
2. 2
   Set the numerator to zero

   Solve x plus 2 equals zero.

   $$x+2=0\Rightarrow x=-2$$

Independent check: At negative 2 the numerator is zero and the denominator is negative 5.

1. 1
   Write standard form

   Expand and subtract 40.

   $$x^2+3x-40=0$$
2. 2
   Factor and interpret

   Use factors 8 and negative 5, then keep the positive length.

   $$(x+8)(x-5)=0\Rightarrow x=5$$

Independent check: A width of 5 and length of 8 have area 40.

1. 1
   Identify the zero factors

   The variable factors are x and x minus 5.

   $$x=0\quad\text{or}\quad x-5=0$$
2. 2
   Solve

   Isolate x in the second equation.

   $$x\in\{0,5\}$$

Independent check: Each value makes one variable factor zero in the original product.