Online calculator for solving linear and quadratic equations with step-by-step explanation
Our calculator allows you to solve linear and quadratic equations online with detailed step-by-step explanations. Simply enter the equation or coefficients and the calculator will instantly find the solution. Ideal for students, schoolchildren and anyone studying mathematics.
2x + 5 = 15
Ответ: x = 5
2x = 15 - 5 = 10, x = 10/2 = 5
3x - 7 = 14
Ответ: x = 7
3x = 14 + 7 = 21, x = 21/3 = 7
5x + 3 = 3x + 8
Ответ: x = 3
5x - 3x = 8 - 2, 2, 2x = 6, x = 3
x² + 5x + 6 = 0
D = 25 - 24 = 1
x₁ = -2, x₂ = -3
D = 5² - 4×1×6 = 1 > 0, two roots
x² - 4x + 4 = 0
D = 16 - 16 = 0
x = 2
D = (-4)² - 4×1×4 = 0, one root
x² + 2x + 5 = 0
D = 4 - 20 = -16
No real roots
D = 2² - 4×1×5 = -16 < 0, complex roots
The linear equation is ax + b = 0, where a and b are known numbers and x is an unknown quantity. To solve, you need to express x.
ax + b = 0 → x = -b/a
Example: 2x + 5 = 15 → 2x = 10 → x = 5
The quadratic equation has the form ax² + bx + c = 0. Solved through the discriminant: D = b² - 4ac. The number of roots depends on the discriminant value.
x = (-b ± √D) / (2a)
D = b² - 4ac
Two roots
D > 0
One root
D = 0
No real roots
D < 0
The roots of an equation are the values of a variable at which the equation turns into a true numerical equality. Depending on the type of equation, there may be a different number of roots.
Our online equation calculator provides fast and accurate solutions with detailed explanations of each step.
Instantly solve any linear and quadratic equations
Detailed step-by-step explanation of the solution
Determination of the number of roots and discriminant
Support 4 languages: Russian, English, Spanish, German
Two input modes: text and coefficients
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Enter the equation in the format '2x + 5 = 15' or 'x² + 2x + 1 = 0' and click the 'Solve' button. The calculator will automatically detect the type of equation and provide a solution with step-by-step explanation.
For the quadratic equation ax² + bx + c = 0, first calculate the discriminant D = b² - 4ac. If D > 0, the equation has two roots, if D = 0 - one root, if D < 0 - there are no real roots.
A linear equation has one solution (if a ≠ 0). A quadratic equation can have 0, 1, or 2 solutions depending on the discriminant. The calculator automatically determines the number of solutions.
Linear equations of the form ax + b = 0 are solved by moving terms: ax = -b, then x = -b/a. For example, 3x + 6 = 0 → 3x = -6 → x = -2.
The discriminant D = b² - 4ac determines the number of roots of the quadratic equation. For D > 0 - two roots, D = 0 - one root, D < 0 - no real roots.
By Vieta's theorem: x₁ + x₂ = -b/a and x₁ × x₂ = c/a. This works for reduced quadratic equations of the form x² + px + q = 0.
Substitute the found value into the original equation. If the left side is equal to the right, the solution is correct. For example, for x = 5 in the equation 2x + 5 = 15: 2×5 + 5 = 15 ✓
If you get a contradiction (for example, 0 = 5), the equation has no solutions. If we get the identity (0 = 0), the equation has infinitely many solutions.
Multiply both sides of the equation by the common denominator to eliminate fractions. Then solve as a normal equation.
Use substitution, addition or graphical methods. Our calculator will show step-by-step solutions for each equation separately.
The root of an equation is the value of the variable at which the equation becomes a true numerical equation. For example, the root of the equation 2x + 3 = 7 is x = 2.
Expand the module by definition: |x| = x for x ≥ 0 and |x| = -x for x < 0. Consider both cases separately.
Raise both sides of the equation to a power to get rid of the root. Be sure to check the resulting roots by substituting them into the original equation.
Reduce to the same base or use logarithms. For example, 2ˣ = 8 → 2ˣ = 2³ → x = 3.
Use the properties of logarithms and the definition: logₐ(b) = c means aᶜ = b. Check the scope.
Use basic trigonometric identities and reduction formulas. Consider the periodicity of trigonometric functions.
Equivalent equations have the same roots. They are obtained from each other using equivalent transformations: addition, subtraction, multiplication by a non-zero number.
Consider different cases of parameter values. For each case, solve the equation separately and indicate conditions on the parameter.
Try factoring, using a change of variable, or special methods (such as Horner's scheme for polynomials).
Express one variable in terms of another or solve as a system of equations. Graphically, it is a straight line or curve on the coordinate plane.
VA is the range of permissible values of a variable for which the expression makes sense. For example, for √x you need x ≥ 0, for 1/x you need x ≠ 0.
Raise both sides to the appropriate power to get rid of the radical. Be sure to check the resulting roots.
Find the zeros of the function, divide the number line into intervals, and determine the sign of the function on each interval.
Graph the left and right sides of the equation. The intersection points of the graphs give the roots of the equation.
Extraneous roots appear when raising an equation to an even power or when multiplying by an expression with a variable. They need to be weeded out by verification.
Enter a new variable instead of a complex expression. Solve the equation for the new variable, then return to the original variable.
Express one variable in terms of another from one equation and substitute it into the second equation. Get an equation with one variable.
Add or subtract equations until one of the variables disappears. Get an equation with one variable.
Enter the equation into our online calculator in text format or using coefficients. Get an instant solution with step-by-step explanation.
Learn the main types of equations: linear, quadratic, rational, irrational. Practice with our calculator with detailed explanations of each step of the solution.
The online equation solving calculator at Calc1.ru supports Russian, English, Spanish and German. Solve linear and quadratic equations with detailed explanations for free!