Build a Quadratic Equation from Roots or a Model

Construct an equation from two roots

If the roots are 2 and 5, the product (x − 2)(x − 5) must vanish at each root. Expanding gives x² − 7x + 10 = 0. Substituting two gives 4 − 14 + 10 = 0, and substituting five gives 25 − 35 + 10 = 0. The signs inside the factors are opposite the root values. For a root of −3, the corresponding factor would be x + 3.

Distinguish roots from a unique function

Multiplying the entire polynomial by a nonzero constant preserves its roots. Thus x² − 7x + 10 = 0 and 4x² − 28x + 40 = 0 have the same solutions. But the functions y = (x − 2)(x − 5) and y = 4(x − 2)(x − 5) have different vertical scales. A leading coefficient or a further point is needed to choose one function. Roots alone determine a family unless a convention such as a monic leading coefficient is specified.

Build a quadratic from a rectangle model

Suppose a rectangle has width x and length x + 3, and area 40. Multiplication gives x(x + 3) = 40, or x² + 3x − 40 = 0. Factoring yields (x + 8)(x − 5) = 0, with roots −8 and 5. The width must be positive, so the valid geometric choice is x = 5 and length 8. Both roots solve the polynomial, but only one satisfies the original dimension requirement.

Put the equation into standard form

Standard form is ax² + bx + c = 0 with a nonzero. Moving all terms to one side makes the coefficients visible for a quadratic-formula tool. For 2x² + 3 = 7x, the standard form is 2x² − 7x + 3 = 0, so a = 2, b = −7 and c = 3. Entering b = 7 would solve another equation. If the x² terms cancel completely, the result is linear or constant rather than quadratic, even if the unsimplified input originally contained squares.

Keep expressions, equations and solutions distinct

An expression such as 3x + 2 describes a value that depends on x. An equation such as 3x + 2 = 14 asserts equality and can be solved for values that make it true. Simplifying an expression does not automatically find a solution because there may be no equality to satisfy. Likewise, a formula containing several variables expresses a relationship rather than supplying all their values. Identify whether the question asks you to simplify, evaluate, rearrange, solve or construct a model before selecting the calculator input.

Preserve equivalence and check the original statement

When solving an equation, apply the same permitted operation to both sides. Distribute a multiplier to every term inside its parentheses, preserve negative signs, and do not divide by an expression that might be zero without treating that case separately. Substitute each candidate into the original equation, not only the last transformed version. Squaring, cancelling or multiplying by a variable denominator can alter the set of allowed values. An answer that satisfies a rearranged equation can still violate the original domain or the physical meaning of a length or count.

Match the local calculator to its supported task

The local equation forms solve linear and quadratic polynomial equations in one variable named x. They are not general symbolic algebra systems, multi-variable formula rearrangers or inequality solvers. Use explicit parentheses for grouped numerators and denominators and use an equals sign when entering an equation so the intended sides are clear. For a geometric or applied problem, derive the equation first and keep the relevant units and constraints in the written working. A numerical solution does not by itself explain which original measurement or variable that number represents.

Use a vertex when that is the given information

A parabola with vertex (h,k) can be written y = a(x − h)² + k. The vertex alone still leaves the nonzero scale a undetermined. If the vertex is (1,−4) and the graph passes through (3,4), then 4 = 4a − 4, giving a = 2. Expanding y = 2(x − 1)² − 4 yields y = 2x² − 4x − 2.

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