ACT Math · Lesson 3 of 5
Equations, quadratics, and sequences
Solve linear and quadratic equations by a method that matches the form, and recognize arithmetic and geometric patterns.
Equations and quadratics
A linear equation has one solution unless it simplifies to a statement that is always true or always false. A quadratic can have two, one, or no real solutions. Factoring, completing the square, or the quadratic formula are tools; the fastest one depends on whether the expression factors cleanly.
If (x − 2)(x + 5) = 0, then x = 2 or x = −5. That conclusion uses the fact that a product is zero only when a factor is zero. It does not apply to (x − 2)(x + 5) = 6 until you rewrite the equation as something equal to zero.
Sequences
An arithmetic sequence adds the same difference. Starting at 7 with difference 4 gives 7, 11, 15, 19. The nth term can be written 7 + (n − 1) × 4 if the first term is term 1.
A geometric sequence multiplies by the same ratio. Starting at 3 with ratio 2 gives 3, 6, 12, 24. Check the first term and the index carefully: using n instead of n − 1 shifts every term.
Worked example
Solve x² − 5x + 6 = 0.
- Look for two numbers that multiply to 6 and add to −5: −2 and −3.
- Factor: (x − 2)(x − 3) = 0.
- The solutions are x = 2 and x = 3. Each makes the original equation true.
Why this works. Factor only after the quadratic equals zero, then set each factor equal to zero.
Check your understanding
- Choose factoring or another method after looking at the numbers.
- Test both candidate solutions in the original equation.
- Say whether a sequence adds a difference or multiplies by a ratio.