Test fractions, factors, and root relationships

1. 1
   Clear fractions

   Multiply the full equation by 4.

   $$2x-(x-3)=20$$
2. 2
   Combine and solve

   Distribute the negative sign and isolate x.

   $$x+3=20\Rightarrow x=17$$

Independent check: Substitution gives 17/2 minus 14/4, which equals 5.

1. 1
   Expand

   Distribute 3 across both terms.

   $$6x+3=6x-4$$
2. 2
   Classify

   Subtract 6x and inspect the remaining statement.

   $$3=-4\Rightarrow\varnothing$$

Independent check: For any x, the left side is 7 greater than the right side.

1. 1
   Match the pattern

   The middle term is negative twice 3a times 4.

   $$(3a)^2-2(3a)(4)+4^2$$
2. 2
   Write the repeated factor

   Use one squared binomial.

   $$(3a-4)^2$$

Independent check: Expansion returns 9a squared minus 24a plus 16.

1. 1
   Take out x

   Both terms share one factor of x.

   $$x(x^2-9)$$
2. 2
   Use the difference of squares

   Nine is 3 squared.

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

Independent check: The conjugate pair multiplies to x squared minus 9, then x restores the cubic.

1. 1
   Factor

   The cross terms negative 6x and positive 4x combine to negative 2x.

   $$(3x+4)(x-2)=0$$
2. 2
   Solve each factor

   Set both linear factors equal to zero.

   $$x=-\frac43\quad\text{or}\quad x=2$$

Independent check: The roots have sum 2/3 and product negative 8/3, matching the coefficient ratios.

1. 1
   Compute the discriminant

   Substitute a = 5, b = 2, and c = 1.

   $$\Delta=2^2-4(5)(1)=-16$$
2. 2
   Interpret the sign

   A negative discriminant gives no real roots.

   $$\Delta<0\Rightarrow0\text{ real solutions}$$

Independent check: Completing the square gives 5(x + 1/5) squared plus 4/5, which is always positive for real x.

1. 1
   Expand both squares

   Use the square of a binomial.

   $$(x^2+4x+4)-(x^2-4x+4)$$
2. 2
   Combine

   Distribute the subtraction and cancel like terms.

   $$8x$$

Independent check: The difference-of-squares identity also gives [(x + 2) - (x - 2)][(x + 2) + (x - 2)] = 4(2x).

1. 1
   Record the restriction

   The denominator cannot be zero.

   $$x\ne1$$
2. 2
   Clear the denominator

   Multiply both sides by x minus 1 and solve.

   $$3=x-1\Rightarrow x=4$$

Independent check: The original left side is 3 divided by 3, which equals 1.

1. 1
   Identify the coefficients

   Here a = 2, b = negative 7, and c = 3.

   $$a=2,\quad b=-7,\quad c=3$$
2. 2
   Apply the relationships

   Use the coefficient ratios.

   $$r_1+r_2=-\frac ba=\frac72,\qquad r_1r_2=\frac ca=\frac32$$

Independent check: Factoring as (2x - 1)(x - 3) gives roots 1/2 and 3, with the stated sum and product.

1. 1
   Form and factor the quadratic

   Expand and set the equation equal to zero.

   $$n^2+n-72=(n+9)(n-8)=0$$
2. 2
   Interpret the roots

   Only the positive root begins a pair of positive integers.

   $$n=8$$

Independent check: Eight and nine are consecutive and their product is 72.