Linear Equations
The equation a x plus b equals zero has the solution x equals minus b over a. For two unknowns, use elimination or substitution.
Quadratic Equations
For a x squared plus b x plus c equals zero, the roots are minus b plus or minus the square root of b squared minus 4ac, all divided by 2a. The discriminant is D equals b squared minus 4ac. If D is positive there are two distinct real roots. If D is zero the roots are equal. If D is negative there are no real roots. The sum of the roots is minus b over a and the product is c over a. If the roots are alpha and beta, the equation is x squared minus the sum times x plus the product equals zero.
Inequalities
Adding or subtracting the same number on both sides keeps the sign. Multiplying or dividing by a negative number reverses the sign.
Word Problems
Let the unknown be x, translate the conditions into an equation, solve, and check. Consecutive numbers are n, n plus 1, n plus 2. Consecutive even or odd numbers are n, n plus 2, n plus 4.
Shortcuts
Equal roots need D equal to zero. One root zero means c is zero. Roots that are negatives of each other mean b is zero. Roots that are reciprocals mean a equals c. When the sum s and product p are known, x squared plus y squared is s squared minus 2p.
Worked Examples
Example 1: Solve x squared minus 5x plus 6 equals zero.
Solution: It factors as x minus 2 times x minus 3, so x is 2 or 3.
Example 2: Find the sum and product of the roots of x squared minus 7x plus 10 equals zero.
Solution: The sum is 7 and the product is 10.
Example 3: Solve x plus y equals 10 and x minus y equals 2.
Solution: Adding gives 2x equals 12, so x is 6 and y is 4.
Quick Check
Question 1: What does D greater than zero mean?
Answer: Two distinct real roots.
Question 2: What happens to an inequality when multiplied by a negative number?
Answer: The sign reverses.
Question 3: What is the sum of the roots of a x squared plus b x plus c?
Answer: Minus b over a.