College Algebra with the Math Sorcerer

Learn that rational equations are equations with fractions, solve by clearing fractions, and check undefined values when denominators become zero, like x cannot be 0, 2, or -2.

Practice adding and subtracting complex numbers by distributing the negative and combining real and imaginary parts; for example, 3 + 2i - (-4 - 6i) simplifies to 7 + 8i.

Distribute the negative, recognize the imaginary unit i, and combine like terms to add and subtract complex numbers, yielding the real part four and imaginary part five i.

Multiply complex numbers by foiling, then apply the conjugate trick to simplify a product by squaring the real and imaginary parts, yielding 29 in the example.

Multiply complex numbers using foil, apply i squared equals minus one, and use a conjugate shortcut to simplify to fifteen.

Learn to divide complex numbers in college algebra by multiplying by the conjugate, applying foil, and expressing the result as (a^2+b^2) in the denominator with a+bi form.

Factor the quadratic x^2 + 4x + 3 into (x+1)(x+3) and apply the zero-product property to solve for x, yielding x = -1 or x = -3.

Use completing the square to solve the quadratic x^2+4x-7=0 by isolating x-terms, forming a perfect square, and obtaining x = -2 ± sqrt(11).

Complete the square to solve the quadratic: get x^2 - x = 1/8, form a perfect square, and apply plus/minus roots to find x = 1/2 ± sqrt(3/8).

Apply the quadratic formula to solve a quadratic by identifying a, b, and c, computing the discriminant, and simplifying to roots 1 and 1/2.

Solve radical equations by isolating the square root, squaring both sides, and solving the resulting quadratic, then verify solutions in the original equation using the zero product principle.

Learn to solve equations with square roots by squaring both sides, factor the resulting quadratic, and verify that x = 8 is the valid root.

Isolate one square root, square both sides, and use the (a−b)^2 expansion to solve equations with two square roots. Verify the original equation to confirm or expose no solution.

Solve for x by isolating square root terms, square both sides, apply foiling, obtain x equals twenty nine, and verify by plugging back to confirm the original equation.

Eliminate the denominator in rational exponents by raising both sides to the corresponding power. Then take the appropriate root to solve for x.

Master substitution techniques to solve higher-degree equations by letting u = x^2, factor x^4 − 13x^2 + 36, and compare with faster direct factoring using difference of squares.

In college algebra, factor the equation with negative exponents and rewrite x^-1 as 1/x. Solve each factor equal to zero to find x = -3/2 and x = -1.

Isolate the absolute value and drop it to form plus or minus cases, solving |x|=3, |x-1|=3 (x=4 or x=-2), and |x-8|=7 (x=15 or x=1).

Solve absolute value equations by dropping the absolute values to form two linear equations, x - 4 = 7x + 12 and x - 4 = -(7x + 12), yielding x = -8/3 and x = -1.

Master solving absolute value inequalities by using the |x|<a and |x|>a rules, converting to intervals, and recognizing unions for greater-than cases.

Solve an absolute value inequality by interpreting |x-4| as distance from 4, show that -3 < x-4 < 3, then add 4 to get 1 < x < 7.

Divide both sides by three to get |x+4| less than three, interpret the absolute value as distance from zero, and conclude that x lies between -7 and -1.