Equation Solver
Solve linear, quadratic, cubic, and polynomial equations with step-by-step solutions. Enter values for instant results with step-by-step formulas.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Equation Solver
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Formula: Quadratic: x = (-b +/- sqrt(b^2 - 4ac)) / 2a
Additional inputs: d (const).
Worked example — x = 3 and x = 2 (sum of roots = 5, product = 6)
Formula
Quadratic: x = (-b +/- sqrt(b^2 - 4ac)) / 2a
The quadratic formula solves ax^2 + bx + c = 0. For linear equations ax + b = 0, x = -b/a. For cubic equations, the Rational Root Theorem and synthetic division reduce the problem to a quadratic. The discriminant (b^2 - 4ac) determines whether roots are real or complex.
Worked Examples
Example 1: Solve a Quadratic Equation
Problem:Solve x^2 - 5x + 6 = 0
Solution:Using the quadratic formula: a=1, b=-5, c=6 Discriminant = (-5)^2 - 4(1)(6) = 25 - 24 = 1 Since discriminant > 0, two real roots exist x = (5 + sqrt(1)) / 2 = 6/2 = 3 x = (5 - sqrt(1)) / 2 = 4/2 = 2 Verification: 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0 Verification: 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0
Result:x = 3 and x = 2 (sum of roots = 5, product = 6)
Example 2: Solve a Cubic Equation
Problem:Solve x^3 - 6x^2 + 11x - 6 = 0
Solution:Try rational roots (factors of 6): x=1, x=2, x=3 Test x=1: 1 - 6 + 11 - 6 = 0 (root found!) Synthetic division by (x-1): x^2 - 5x + 6 = 0 Solve quadratic: discriminant = 25 - 24 = 1 x = (5+1)/2 = 3 and x = (5-1)/2 = 2 All three roots: x = 1, x = 2, x = 3
Result:x = 1, x = 2, and x = 3
Frequently Asked Questions
What is a linear equation and how do I solve it?
A linear equation is an equation where the variable appears only to the first power, taking the general form ax + b = 0 where a and b are constants. To solve a linear equation, isolate the variable x by performing inverse operations on both sides. Subtract b from both sides to get ax = -b, then divide both sides by a to get x = -b/a. Linear equations always have exactly one solution unless a equals zero, in which case the equation is either an identity (true for all x when b also equals zero) or a contradiction (no solution when b is not zero). Linear equations represent straight lines when graphed and are the foundation for more complex equation types.
What does the discriminant tell us about an equation?
The discriminant is the expression under the square root in the quadratic formula, calculated as b squared minus 4ac. It reveals crucial information about the nature and number of solutions without actually solving the equation. A positive discriminant means the equation has two distinct real roots and the parabola crosses the x-axis at two points. A discriminant of zero means there is exactly one real root (a repeated root) and the parabola touches the x-axis at its vertex. A negative discriminant means there are no real roots, only two complex conjugate roots, and the parabola does not intersect the x-axis at all. The discriminant is also useful for analyzing conic sections and higher-degree polynomials.
How do I solve a cubic equation?
Solving cubic equations of the form ax cubed plus bx squared plus cx plus d equals zero typically involves finding one rational root using the Rational Root Theorem, then reducing to a quadratic equation through synthetic division or polynomial long division. The Rational Root Theorem states that any rational root p over q must have p as a factor of the constant term d and q as a factor of the leading coefficient a. Once you find one root, divide the cubic by the factor (x minus that root) to get a quadratic, which you solve using the quadratic formula. If no rational roots exist, numerical methods like Newton-Raphson iteration or Cardano formula for the depressed cubic can find the roots.
What is the relationship between roots and coefficients?
Vieta formulas describe the relationship between the roots of a polynomial and its coefficients. For a quadratic ax squared plus bx plus c equals zero with roots r1 and r2, the sum of roots equals negative b over a and the product of roots equals c over a. For a cubic with roots r1, r2, and r3, the sum equals negative b over a, the sum of products taken two at a time equals c over a, and the product of all three roots equals negative d over a. These relationships are useful for checking solutions, constructing equations from known roots, and understanding how changing coefficients affects the roots. They work for polynomials of any degree.
What is completing the square and when is it useful?
Completing the square is an algebraic technique that rewrites a quadratic expression ax squared plus bx plus c in the form a times the quantity x plus h squared plus k, where h equals negative b over 2a and k equals c minus b squared over 4a. This form immediately reveals the vertex of the parabola at the point (negative h, k) and makes it easy to solve the equation by taking square roots. Completing the square is essential for deriving the quadratic formula, converting quadratic equations to vertex form for graphing, and in calculus for integrating certain expressions. It is also used in multivariable calculus and optimization problems to identify minimum and maximum values.
Can Equation Solver solve equations with fractions or decimals?
Yes, Equation Solver accepts any real number coefficients including fractions expressed as decimals. Enter 0.5 for one-half, 0.333 for one-third, 1.5 for three-halves, and so on. The calculator processes all arithmetic using standard floating-point numbers, which provides accuracy to approximately 15 significant digits for most calculations. For equations with fractional coefficients, you can also multiply all terms by the least common denominator to convert to integer coefficients before entering them. For example, instead of entering 0.5x squared plus 1.5x plus 1 equals zero, you could enter x squared plus 3x plus 2 equals zero by multiplying everything by 2.
What is the vertex of a parabola and how do I find it?
The vertex of a parabola described by the equation y equals ax squared plus bx plus c is the point where the parabola changes direction, representing either the minimum value (when a is positive, opening upward) or the maximum value (when a is negative, opening downward). The x-coordinate of the vertex is negative b divided by 2a, and the y-coordinate is found by substituting this x value back into the equation, giving c minus b squared over 4a. The vertex is also the midpoint between the two roots of the quadratic equation if real roots exist. The axis of symmetry of the parabola is the vertical line passing through the vertex at x equals negative b over 2a.
How do I verify that my solutions are correct?
To verify solutions, substitute each root back into the original equation and confirm that both sides are equal. For the equation x squared minus 5x plus 6 equals zero with solutions x equals 2 and x equals 3, check that 2 squared minus 5 times 2 plus 6 equals 4 minus 10 plus 6 equals 0, and that 3 squared minus 5 times 3 plus 6 equals 9 minus 15 plus 6 equals 0. For equations with irrational or complex roots, the substitution should yield a result very close to zero, typically within rounding error of about one billionth. You can also verify using Vieta formulas by checking that the sum and product of your roots match the expected values from the coefficients.
What is the difference between an equation and an expression?
An equation contains an equals sign and states that two expressions are equal, such as 2x plus 3 equals 7, and it can be solved to find specific values of the variable that make the statement true. An expression is a mathematical phrase without an equals sign, such as 2x plus 3, and it cannot be solved but can be simplified or evaluated for specific values of x. Equations have solutions (specific values), while expressions have values that depend on the variable. You can simplify expressions by combining like terms and applying operations, but solving requires an equation with two sides. Equation Solver works with equations, finding the values of x that make the left side equal to zero.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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