Use algebraic and/or numeric methods as the
main technique for solving a math problem. Usually when a problem
is solved analytically, no graphing calculator is used.
Step 1: Factor the left side by finding two numbers that multiply to 6 and add to −5.
(x−2)(x−3)=0
Step 2: Set each factor equal to zero and solve.
x−2=0⟹x=2orx−3=0⟹x=3
Answer: x = 2 or x = 3. These are exact solutions found entirely through algebra — no graph or calculator needed.
Why It Matters
Analytical solutions give you exact answers, not approximations. When you solve graphically, you might read an intersection point as x≈2.98 when the true answer is x=3. Many standardized tests and college courses require analytical methods to demonstrate that you understand the underlying algebra.
Common Mistakes
Mistake: Confusing an analytical solution with a numerical approximation.
Correction: An analytical solution is an exact, symbolic answer (like x = 3 or x = √2). If you use a calculator to get a decimal estimate (like x ≈ 1.414), that is a numerical method, not a purely analytical one.
Related Terms
Solve Graphically — Alternative approach using graphs instead of algebra
Algebra — The primary toolset used in analytical solving
Equation — The type of statement most often solved analytically
Factoring — A common analytical technique for polynomials
Quadratic Formula — An analytical formula for solving quadratic equations
Mathwords