Expression Simplifier

An expression is a piece of algebra with no equals sign: $3(2x - 5) - 4(x + 1)$. An equation has one: $3x = 12$.

That difference decides what you are allowed to do. An equation is a balance, so you can subtract the same thing from both sides. An expression has no sides — every move must leave its value unchanged for every $x$. So you may rewrite $3x + 2x$ as $5x$, but you may never "divide both sides by 3", because there is nothing to divide.

Simplifying therefore means one thing: rewrite the expression in the shortest equivalent form. The finished form has no brackets left, and no two terms that could still be combined. "Solve" the same expression and there is nothing to find — an expression has no answer until you supply a value for the variable.

1. Distribute. Multiply the term outside the brackets by every term inside:

$3(2x - 5) = 6x - 15$

A minus sign in front of a bracket is a $-1$ being distributed, so it flips every sign inside: $-4(x + 1) = -4x - 4$.

2. Collect like terms. Terms are like only when their variable parts are identical, letters and exponents. So $6x$ and $-4x$ combine to $2x$; $3x$ and $3x^2$ never do, and neither do $5a$ and $5b$. Add the coefficients and keep the variable part unchanged.

Evaluating is a separate job: substitute the number for the variable and follow the order of operations. Put brackets around any negative substitution — $2x^2$ at $x = -3$ is $2(-3)^2 = 2 \times 9 = 18$, not $-18$. Simplify first when you can; there is less to evaluate.

• Distributing to the first term only. $3(2x - 5)$ is $6x - 15$, not $6x - 5$.
• Losing the sign after a subtracted bracket. $7 - 2(x - 6) = 7 - 2x + 12 = 19 - 2x$. The $-2$ times $-6$ is $+12$.
• Combining unlike terms. $3a + 4b$ stays as it is; there is no shorter form.
• Adding the exponents when collecting. $x^2 + x^2 = 2x^2$, not $x^4$. Exponents add when you multiply powers, not when you add terms.
• Multiplying the exponent by the coefficient. In $2x^3$ only the $x$ is cubed; at $x = 2$ that is $2 \times 8 = 16$, not $6^3$.
• Missing brackets on a negative substitution. $-x^2$ at $x = -3$ is $-9$, while $(-x)^2$ is $9$.