Algebra Cheat Sheet: Solving Linear & Quadratic Equations

Posted on May 14, 2024 in Mathematics

Section 1

Chapter 1: Solving Linear Equations Cheat Sheet

Objective 1: Solve a Linear Equation

Concepts and Rules:

• A linear equation is in the form of ax + b = c, where a, b, and c are real numbers.
• The goal is to isolate the variable (x) on one side of the equation.
• Balancing: Perform the same operation on both sides to maintain equality.

Step-by-Step Solving Process:

1. Start with the equation in the form ax + b = c.
2. Use inverse operations to isolate x:
   • To remove/add a constant (b): Perform the opposite operation (addition or subtraction) on both sides.
   • To remove/multiply by a coefficient (a): Perform the opposite operation (division or multiplication) on both sides.
3. Simplify both sides of the equation.
4. The solution is the value of x that makes the equation true.

Example 1:

Solve for x: 2x + 3 = 7

• Subtract 3 from both sides: 2x = 4
• Divide by 2 on both sides: x = 2

Objective 2: Solve Equations That Lead to Linear Equations

Concepts and Rules:

• Simplify Complex Equations: Transform complex equations into linear form.
• Equivalent Equations: Equivalent equations have the same solution set.
• Five Methods for Equivalent Equations:
  1. Interchange sides of the equation.
  2. Simplify by combining like terms.
  3. Add or subtract the same expression on both sides.
  4. Multiply or divide both sides by the same nonzero expression.
  5. Factor and set each factor equal to zero (Zero-Product Property).

Step-by-Step Process:

1. Start with a complex equation.
2. Apply one or more of the five methods to simplify the equation into linear form.
3. Solve the simplified linear equation.
4. Ensure that the solutions also satisfy the original complex equation.

Example 2:

Solve the equation: (2y + 1)(y – 1) = (y + 5)(2y – 5)

• Expand both sides: 2y² – y – 1 = 2y² + 5y – 25
• Collect like terms: -y – 1 = 5y – 25
• Move terms to isolate y: -y – 5y = -25 + 1
• Combine like terms: -6y = -24
• Divide by -6: y = 4

1.2 Section

Objective 1: Solve a Quadratic Equation by Factoring

Concepts and Rules:

• A quadratic equation is in the form ax^2 + bx + c = 0, where a, b, and c are real numbers.
• Factoring is the process of rewriting the quadratic equation as the product of two first-degree polynomials.
• The Zero-Product Property states that if the product of two factors is zero, at least one of the factors must be zero.
• Solving a quadratic equation by factoring involves finding the values of x that make each factor equal to zero.

Step-by-Step Solving Process:

1. Write the quadratic equation in standard form: ax^2 + bx + c = 0.
2. Factor the left side of the equation into two first-degree polynomials.
3. Use the Zero-Product Property to set each factor equal to zero.
4. Solve the resulting linear equations for x.
5. The solutions to the quadratic equation are the values of x obtained in step 4.

Objective 2: Solve a Quadratic Equation Using the Square Root Method

Concepts and Rules:

• The Square Root Method is used to solve quadratic equations in the form x^2 = p, where p > 0.
• If x^2 = p, then x = ±√p, which means there are two solutions: x = √p and x = -√p.

Step-by-Step Solving Process:

1. Write the quadratic equation in the form x^2 = p.
2. Isolate x^2 on one side of the equation.
3. Apply the Square Root Method: x = ±√p.
4. Express the solutions with both the positive and negative square roots.

Objective 3: Solve a Quadratic Equation by Completing the Square

Concepts and Rules:

• Completing the square is a method to solve any quadratic equation in the form ax^2 + bx + c = 0.
• The goal is to rewrite the equation as a perfect square trinomial on one side.
• To complete the square, add and subtract (b/2)^2 inside the equation.

Step-by-Step Solving Process:

1. Write the quadratic equation in the form ax^2 + bx + c = 0.
2. If the coefficient of x^2 is not 1, divide the entire equation by that coefficient.
3. Move the constant term (c) to the other side.
4. Complete the square by adding and subtracting (b/2)^2 inside the equation.
5. Rewrite the left side as a perfect square trinomial.
6. Take the square root of both sides.
7. Solve for x.
8. Express the solutions.

Objective 4: Solve a Quadratic Equation Using the Quadratic Formula

Concepts and Rules:

• The Quadratic Formula is a universal method for solving any quadratic equation in the form ax^2 + bx + c = 0.
• The discriminant, b^2 – 4ac, determines the nature and number of solutions.
• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is one real solution (a repeated root).
• If the discriminant is negative, there are no real solutions.

Step-by-Step Solving Process:

1. Write the quadratic equation in standard form: ax^2 + bx + c = 0.
2. Identify the coefficients a, b, and c.
3. Calculate the discriminant: b^2 – 4ac.
4. Determine the number and nature of solutions based on the discriminant.
5. If there are real solutions, use the Quadratic Formula: x = (-b ± √(b^2 – 4ac)) / (2a).
6. Plug in the values of a, b, and c into the formula.
7. Simplify and calculate the solutions.

Remember to evaluate the discriminant first to determine the type and number of solutions for a quadratic equation.

1.3 Section

Objective 1: Add, Subtract, Multiply, and Divide Complex Numbers

Complex Number Basics

• A complex number is of the form a+bi, where ‘a’ and ‘b’ are real numbers.
• ‘a’ is the real part, ‘b’ is the imaginary part, and ‘i’ is the imaginary unit (i² = -1).

Equality of Complex Numbers

• Two complex numbers a+bi and c+di are equal if and only if a=c and b=d.

Sum of Complex Numbers

• To add two complex numbers (a+bi) and (c+di), add their real parts and imaginary parts separately: (a+bi) + (c+di) = (a+c) + (b+d)i.

Difference of Complex Numbers

• To subtract two complex numbers (a+bi) and (c+di), subtract their real parts and imaginary parts separately: (a+bi) – (c+di) = (a-c) + (b-d)i.

Multiplying Complex Numbers

• Multiply complex numbers as you would binomials, and remember that i² = -1: (a+bi)(c+di) = (ac – bd) + (ad + bc)i.

Dividing Complex Numbers

• To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator: (a+bi) / (c+di) = [(a+bi)(c-di)] / [(c+di)(c-di)] = [(ac+bd) + (bc-ad)i] / (c² + d²).

Objective 2: Solve Quadratic Equations in the Complex Number System

Quadratic Equations with Negative Discriminants

• Quadratic equations of the form ax² + bx + c = 0, where a ≠ 0 and a, b, and c are real numbers, may not have real solutions if the discriminant (b² – 4ac) is negative.

Principal Square Root of -N

• If N is a positive real number, the principal square root of -N is defined as -N = Ni, where i is the imaginary unit (i² = -1).

Quadratic Formula (Complex Solutions)

• In the complex number system, solutions to the quadratic equation ax² + bx + c = 0 are given by: x = (-b ± √(b² – 4ac)) / (2a).

Character of Solutions

• The character of solutions to a quadratic equation depends on the discriminant:
  • If b² – 4ac > 0, there are two unequal real solutions.
  • If b² – 4ac = 0, there is one repeated real solution (double root).
  • If b² – 4ac < 0, there are two complex solutions (conjugates of each other).

Key Concepts and Rules

Complex Conjugate

• The conjugate of a complex number z = a+bi is z¯ = a-bi. It is found by changing the sign of the imaginary part.

Properties of Complex Conjugates

• z¯¯ = z (conjugate of a conjugate is the number itself).
• (z + w)¯ = z¯ + w¯ (conjugate of a sum is the sum of conjugates).
• (z * w)¯ = z¯ * w¯ (conjugate of a product is the product of conjugates).

Powers of i

• Powers of i follow a repeating pattern: i, -1, -i, 1.

1.4 Section

Objective 1: Solve Radical Equations

Concept: Radical equations are equations that involve radicals (square roots, cube roots, etc.). To solve them, isolate the radical term and eliminate it by raising both sides to a power equal to the index of the radical.

Procedure:

1. Isolate the radical term on one side of the equation.
2. Raise both sides to the power equal to the index of the radical.
3. Solve for the variable.
4. Check for extraneous solutions by verifying them in the original equation.

Objective 2: Solve Equations Quadratic in Form

Concept: Equations that can be transformed into quadratic form are called equations quadratic in form. Substitution is often used to make the equation quadratic, and then it can be solved like a quadratic equation.

Procedure:

1. Recognize that the equation can be transformed into a quadratic form.
2. Make an appropriate substitution (usually involving a new variable) to create a quadratic equation.
3. Solve the quadratic equation.
4. If necessary, solve for the original variable by reversing the substitution.
5. Check for extraneous solutions by verifying them in the original equation.

Objective 3: Solve Equations by Factoring

Concept: Factoring involves expressing an equation as a product of two or more expressions. To solve equations by factoring, identify common factors and apply the zero-product property.

Procedure:

1. Arrange the equation so that one side is set equal to zero.
2. Factor the equation as much as possible, looking for common factors.
3. Set each factor equal to zero (Zero-Product Property).
4. Solve for the variable in each equation obtained from the factors.
5. Check each solution by substituting it into the original equation.

Here’s a summary of these concepts:

Objective 1: Solve Radical Equations

• Isolate the radical term.
• Raise both sides to the power of the radical’s index.
• Solve for the variable.
• Check for extraneous solutions in the original equation.

Objective 2: Solve Equations Quadratic in Form

• Recognize equations that can be transformed into quadratic form.
• Make an appropriate substitution to create a quadratic equation.
• Solve the quadratic equation.
• Reverse the substitution to solve for the original variable if needed.
• Check for extraneous solutions in the original equation.

Objective 3: Solve Equations by Factoring

• Set one side of the equation equal to zero.
• Factor the equation, looking for common factors.
• Apply the Zero-Product Property by setting each factor equal to zero.
• Solve for the variable in each equation.
• Check each solution by substituting it into the original equation.

1.5 Section

Objective 1: Use Interval Notation

Definition of Intervals:

• Open Interval (a, b): All real numbers x for which a < x < b.
• Closed Interval [a, b]: All real numbers x for which a ≤ x ≤ b.
• Half-Open Intervals: [a, b) and (a, b]: Similar to closed and open intervals with one endpoint included and one excluded.
• Infinity (∞): Represents unboundedness in the positive direction.
• Negative Infinity (-∞): Represents unboundedness in the negative direction.

Table 1: Interval Notation

Objective 2: Use Properties of Inequalities

Nonnegative Property:

• For any real number a, a^2 ≥ 0.

Addition Property of Inequalities:

• If a < b, then a + c < b + c.
• If a > b, then a + c > b + c.

Multiplication Properties for Inequalities:

• If a < b and c > 0, then ac < bc.
• If a < b and c < 0, then ac > bc.
• If a > b and c > 0, then ac > bc.
• If a > b and c < 0, then ac < bc.

Reciprocal Properties for Inequalities:

• If a > 0, then 1/a > 0.
• If a < 0, then 1/a < 0.
• If b > a > 0, then 1/a > 1/b > 0.
• If a < b < 0, then 1/b < 1/a < 0.

Objective 3: Solve Inequalities

Procedures That Leave the Inequality Symbol Unchanged:

1. Simplify both sides by combining like terms and eliminating parentheses.
2. Add or subtract the same expression on both sides.
3. Multiply or divide both sides by the same positive expression.

Procedures That Reverse the Sense of the Inequality Symbol:

1. Interchange the two sides of the inequality.
2. Multiply or divide both sides of the inequality by the same negative expression.

Objective 4: Solve Combined Inequalities

Solving Combined Inequalities:

• When dealing with combined inequalities (e.g., -5 < 3x – 2 < 1), you can solve each part separately, applying the same steps to each inequality.
• Shortcut: You can also solve them together as a single inequality by following the same rules and properties.

Application: Ohm’s Law

• Ohm’s law: E = IR, where E is voltage (V), I is current (A), and R is resistance (Ω).
• Given resistance R = 10 Ω and voltage varies from 110V to 120V (inclusive).
• Calculate the current I using E = IR, leading to the range 11 A ≤ I ≤ 12 A.

1.6 Section

Objective 1: Solve Equations Involving Absolute Value

Theorem:

• If |u| = a (where a is a positive number), then u = a or u = -a.
• If a = 0, then |u| = 0 is equivalent to u = 0.
• If a is negative, there is no real solution for |u| = a.

Example 1: Solving Equations Involving Absolute Value

• Given |x + 4| = 13, solve for x:
  • x + 4 = 13 or x + 4 = -13
  • x = 9 or x = -17
  • Solution set: {9, -17}
• Given |2x – 3| + 2 = 7, solve for x:
  • |2x – 3| = 5
  • 2x – 3 = 5 or 2x – 3 = -5
  • 2x = 8 or 2x = -2
  • x = 4 or x = -1
  • Solution set: {4, -1}

Objective 2: Solve Inequalities Involving Absolute Value

Theorem:

• If |u| < a, it is equivalent to -a < u < a.
• If |u| ≤ a, it is equivalent to -a ≤ u ≤ a.

Example 2: Solving an Inequality Involving Absolute Value

• Given |x| < 4, the solution set is (-4, 4).

Objective 3: Solve Inequalities Involving Absolute Value

Theorem:

• If |u| > a, it is equivalent to u < -a or u > a.
• If |u| ≥ a, it is equivalent to u ≤ -a or u ≥ a.

Example 3: Solving an Inequality Involving Absolute Value

• Given |2x + 4| ≤ 3, the solution set is [-7/2, -3/2].

Example 4: Solving an Inequality Involving Absolute Value

• Given |1 – 4x| < 5, the solution set is (-1, 3/2).

Example 5: Solving an Inequality Involving Absolute Value

• Given |x| > 3, the solution set is (-∞, -3) ∪ (3, ∞).

Example 6: Solving an Inequality Involving Absolute Value

• Given |2x – 5| > 3, the solution set is (-∞, 1) ∪ (4, ∞).

Important Notes:

• Be careful not to mix inequality symbols (e.g., writing x < 1 and x > 4 as 1 > x > 4 is incorrect).
• Pay attention to the direction of the inequality symbol when dealing with negative values of a.