Calculate Quadratic Roots and Check the Discriminant

Write the formula with the full denominator

For ax² + bx + c = 0 and a nonzero, x = [−b ± √(b² − 4ac)]/(2a). The entire numerator is divided by 2a. A common input error divides only the square-root part or forgets that −b changes the sign of b. First record a, b and c directly from standard form. If a linear or constant term is absent, use zero for its coefficient rather than leaving the required field blank.

Work an equation with two real roots

For 2x² − 7x + 3 = 0, a = 2, b = −7 and c = 3. The discriminant is 49 − 24 = 25, so x = (7 ± 5)/4. The roots are 3 and 1/2. Check at three: 18 − 21 + 3 = 0. Check at one half: 1/2 − 7/2 + 3 = 0. The two signs in the formula generate separate candidates; keeping only the plus sign would omit a valid root.

Interpret zero or negative discriminants

For x² − 6x + 9 = 0, the discriminant is zero and the repeated root is 3. For x² + 4x + 5 = 0, the discriminant is −4, giving complex roots −2 ± i and no real roots. “No real roots” does not mean no roots in the complex number system. State the requested number domain, especially in a graphing question where real x-intercepts and complex equation solutions serve different purposes.

Use root relationships as an extra check

The roots of a quadratic have sum −b/a and product c/a, counting a repeated root twice. In the first example, 3 + 1/2 = 7/2 and 3×1/2 = 3/2, agreeing with the coefficients. These relationships provide a quick independent check on sign or denominator mistakes. For measured or decimal coefficients, preserve adequate intermediate precision and compare substituted values within a sensible tolerance; displayed rounded roots may leave a small residual even when the unrounded calculation is correct.

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.

Recognize the linear boundary case

If a is zero, the equation bx + c = 0 is linear and should not be sent through a formula that divides by 2a. With b nonzero, its solution is −c/b. If b is also zero, the constant c determines whether the statement is an identity or inconsistent. These cases explain why a zero quadratic coefficient is a meaningful input rejection rather than a missing quadratic root.

Related guides

Build a Quadratic Equation from Roots or a Model

Construct a quadratic from stated roots, convert an area model to standard form, and explain what additional information fixes its scale.

Solve an Equation Online and Verify Its Roots

Enter linear or quadratic equations with clear grouping, compare valid solver outcomes, and verify roots against the original equation and domain.

Find Rectangle Dimensions from Area and Perimeter

Recover rectangle sides from area and perimeter, solve the resulting quadratic, and identify impossible or insufficient measurement combinations.