Worked Examples

Arithmetic with mixed operations

Try 2 + 3 * (4 - 1)^2 in the solver. Reading it correctly means working from the inside out: the parentheses first (4 - 1 = 3), then the exponent (3^2 = 9), then the multiplication (3 * 9 = 27), and finally the addition (2 + 27 = 29). Each of those four operations becomes its own row in the step list, in exactly that order, regardless of the order the operations happen to be written in the original expression.

This example is a good stress test for the order of operations because it uses all four levels of precedence (parentheses, exponents, multiplication, addition) in a single line. Getting any one of the four steps out of order, for instance adding 2 + 3 before evaluating the exponent, produces a different and incorrect final answer.

A linear equation with the variable on both sides

Try 5x - 2 = 2x + 7. Unlike the homepage's basic example, this one has the variable term on both sides of the equals sign. The solver's approach does not change: it moves everything to one side first, giving 5x - 2 - (2x + 7) = 0, then expands and combines like terms. The two x-terms combine into 3x, and the two constants combine into -9, leaving 3x - 9 = 0. From there it isolates the variable exactly as before: 3x = 9, then x = 9 / 3, giving x = 3.

This matters because a common mistake when solving by hand is to move only one of the two x-terms, or to forget to flip a sign when a term crosses the equals sign. Because the solver always works from a single normalized form (everything on one side, set equal to zero) rather than shuffling terms back and forth across an equals sign, this class of sign error cannot happen in its output.

A linear equation with distributed parentheses

Try 3*(x + 2) - 2*x = 5*x - 1. Both sides need to be expanded before like terms can be combined: the left side becomes 3x + 6 - 2x, which simplifies to x + 6, and the right side is already in simple form. Moving everything to one side and expanding gives x + 6 - (5x - 1) = 0, which combines to -4x + 7 = 0. Isolating the variable gives -4x = -7, and dividing both sides by -4 gives x = -7 / -4, which simplifies to x = 7/4 (shown alongside its decimal, 1.75).

This example demonstrates why the solver leans on a real expansion step (via its underlying math library, rather than a hand-written distribution routine) rather than trying to distribute terms with a simple find-and-replace approach: the parentheses here interact with terms on the same side that also need to be combined afterward, and doing that reliably for arbitrary input is exactly the kind of parsing problem a dedicated math library is built to get right.

A quadratic equation with a repeated root

Try x^2 + 6x + 9 = 0. Expanding gives the standard form directly (it is already in that form): a = 1, b = 6, c = 9. The discriminant, b^2 - 4ac, works out to 36 - 36 = 0. A discriminant of exactly zero means there is only one solution rather than two, found with x = -b / (2a): x = -6 / 2 = -3. You can check this by noticing that x^2 + 6x + 9 is a perfect square, (x + 3)^2, which is zero only when x = -3.

A quadratic equation with no real solutions

Try x^2 + 2x + 5 = 0. Here a = 1, b = 2, c = 5, and the discriminant is (2)^2 - 4(1)(5) = 4 - 20 = -16, which is negative. Rather than attempting to take the square root of a negative number and produce a nonsensical result, the solver reports plainly that there are no real solutions. Graphically, this corresponds to a parabola that never crosses the x-axis at all, it stays entirely above it for every value of x.

An equation with infinitely many solutions

Try 2*(x + 3) = 2x + 6. Expanding the left side gives 2x + 6, which is identical to the right side. Moving everything to one side gives 2x + 6 - (2x + 6) = 0, which simplifies all the way down to 0 = 0, a statement that is always true no matter what x is. The solver recognizes this pattern (every variable term cancels out and what remains is true) and reports that the equation holds for every value of x, rather than trying to force a single numeric answer out of an equation that does not have just one.

An equation with no solution at all

Try x + 4 = x + 9. Moving everything to one side gives x + 4 - (x + 9) = 0, and the x-terms cancel completely, leaving -5 = 0, a statement that is never true regardless of x. The solver reports that this equation has no solution. This is a different outcome from the negative-discriminant quadratic case above, but the underlying idea is the same: not every equation you can write down actually has an answer, and recognizing that is as much a part of solving correctly as finding a numeric root when one exists.

Simplifying an expression without solving an equation

If you type an expression with a variable but no equals sign, such as 3*(y + 2) - 2*y - 8, the solver does not try to solve for y (there is nothing to solve, since it is not an equation), it simplifies the expression instead. Expanding gives 3y + 6 - 2y - 8, and combining like terms gives the simplified form, y - 2. This is useful on its own when you have a messy expression partway through a larger problem and just want it cleaned up before continuing by hand.

Reading the discriminant table

A quick reference for the quadratic cases above:

Head back to the math solver to try your own equation, or use the sign of b^2 - 4ac as a quick sanity check before trusting any quadratic answer, your own or a tool's.