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Solve linear, quadratic, and systems of equations with step-by-step work, verification, and discriminant analysis. All calculations run in your browser.
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Verification
A linear equation has the form ax + b = c, where a, b, and c are constants and x is the unknown. The solution process involves isolating x by performing inverse operations on both sides.
Special cases: if a = 0 and b = c, there are infinitely many solutions. If a = 0 and b is not equal to c, there is no solution.
A quadratic equation has the form ax^2 + bx + c = 0. The quadratic formula provides a direct method to find solutions:
The discriminant (b^2 - 4ac) determines root nature:
A system of two linear equations with two unknowns can be solved using elimination or Cramer's rule. The elimination method involves manipulating equations to eliminate one variable, solving for the other, then substituting back.
Systems may have one unique solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (same line).
After finding the solution, substitute the value(s) back into the original equation. If both sides evaluate to the same number, the solution is verified. This is an essential step in algebra to confirm accuracy.
To solve ax + b = c: First, subtract b from both sides to get ax = c - b. Then divide both sides by a to get x = (c - b) / a. Always perform the same operation on both sides to maintain equality.
If a = 0, the equation becomes b = c, which has no variable. If b equals c, any value of x is a solution (infinite solutions). If b does not equal c, there is no solution.
Use the quadratic formula: x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Calculate the discriminant (b^2 - 4ac) first. If it is positive, take its square root and use both + and - to get two solutions. If zero, the +/- part vanishes and there is one solution. If negative, the roots are complex numbers.
The discriminant is the expression b^2 - 4ac under the square root in the quadratic formula. It tells you how many real solutions exist: two if positive, one if zero, or none (only complex) if negative.
Yes. Enter decimal values (e.g., 0.5 for 1/2) or use fractional decimals. The results will be shown as decimals and, where appropriate, exact forms will be displayed for common values.
One method is elimination: multiply one or both equations by constants so that adding or subtracting them eliminates one variable. Solve for the remaining variable, then substitute back into either original equation to find the other variable.
A system has no solution if the equations represent parallel lines (same slope, different intercepts). Algebraically, this occurs when the elimination process produces a contradiction like 0 = 5.
No. All calculations run entirely in your browser. Nothing is sent to a server, and no data is stored.
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Solve ax² + bx + c = 0 with step-by-step solution
To solve a linear equation ax + b = c, first subtract b from both sides to get ax = c - b, then divide both sides by a to get x = (c - b) / a. Special cases: if a = 0 and c - b = 0, there are infinite solutions; if a = 0 and c - b is not 0, there is no solution.
Use the quadratic formula: x = (-b +/- sqrt(b^2 - 4ac)) / (2a). First calculate the discriminant (b^2 - 4ac). If positive, there are 2 real roots. If zero, there is 1 repeated root. If negative, there are 2 complex roots.
The discriminant is b^2 - 4ac from the quadratic formula. It determines the nature of the roots: positive means 2 distinct real roots, zero means 1 repeated real root, negative means 2 complex conjugate roots.
One method is elimination: multiply equations to make coefficients of one variable equal, then add or subtract to eliminate that variable. Solve for the remaining variable, then substitute back to find the other.
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