Equation Solver: How To Solve Any Type Of Equation Step By Step

Every equation is a statement that two things are equal. Solving it means finding the value, or values, that keep that statement true. This guide walks through each common type of equation with a worked example, explains when a solver's answer is fake (an "extraneous" solution), and shows how to check any result. Every example below was verified with a computer algebra system.

Note: Calc Solver (calc-solver.co) is an independent guide site and is not affiliated with any solver mentioned here.

What An Equation Solver Does

An equation solver isolates the unknown variable by doing the same thing to both sides of the equation until the variable stands alone. It then reports every value that makes the equation true, or says there is no solution.

Think of an equation as a balanced scale. Whatever you add, subtract, multiply or divide on one side, you must do on the other. A solver follows those same rules, just faster and without arithmetic slips.

Equations can have:

• One solution: 2x + 1 = 7 has only x = 3.
• Several solutions: x² = 9 has x = 3 and x = −3.
• No solution: x + 1 = x + 2 is never true.
• Infinitely many solutions: 2x + 4 = 2(x + 2) is true for every x.

How To Type Equations Into A Solver

Correct input is the most important step, because a solver solves exactly what you type. Use this format with most online equation solvers.

Always include the equals sign. Without it, many tools will simplify the expression instead of solving it.

Linear Equations

A linear equation has the variable only to the first power, like 3x + 2 = 11. Solve it by moving variable terms to one side and numbers to the other, then dividing.

Linear Equations With Fractions

When an equation has fractions, multiply every term by the least common denominator first to clear them.

Example: x/3 + 5 = 2x − 5

1. Multiply every term by 3: x + 15 = 6x − 15
2. Subtract x from both sides: 15 = 5x − 15
3. Add 15 to both sides: 30 = 5x
4. Divide by 5: x = 6

Check: 6/3 + 5 = 7 and 2(6) − 5 = 7. Both sides match.

No Solution And Infinitely Many Solutions

If the variable disappears while solving, look at what is left. A false statement means no solution; a true statement means every number works.

Quadratic Equations

A quadratic equation has x² as its highest power, written in standard form as ax² + bx + c = 0. It can have two, one or no real solutions.

Choosing A Method

There are three main ways to solve a quadratic. Pick the fastest one for the numbers you have.

Using The Quadratic Formula

The quadratic formula always works: x = (−b ± √(b² − 4ac)) ÷ 2a.

Example: x² + 4x − 1 = 0 (a = 1, b = 4, c = −1)

1. Discriminant: b² − 4ac = 16 − 4(1)(−1) = 20
2. Plug in: x = (−4 ± √20) ÷ 2
3. Simplify √20 = 2√5: x = (−4 ± 2√5) ÷ 2
4. Answer: x = −2 + √5 ≈ 0.236 or x = −2 − √5 ≈ −4.236

What The Discriminant Tells You

The discriminant, b² − 4ac, tells you how many solutions to expect before you finish. According to Math is Fun, a positive value gives two real solutions, zero gives one, and a negative value gives a pair of complex solutions.

Example: x² + 2x + 5 = 0. The discriminant is 4 − 20 = −16, so there are no real solutions. The complex solutions are x = −1 + 2i and x = −1 − 2i. Many solvers show "no real solution" unless you turn on complex answers.

Cubic And Higher Polynomial Equations

A polynomial equation of degree 3 or higher can have up to as many solutions as its degree. A cubic has up to 3 real solutions.

Example: x³ − 6x² + 11x − 6 = 0

1. Try small whole numbers that divide the constant 6 (±1, ±2, ±3, ±6). x = 1 works: 1 − 6 + 11 − 6 = 0.
2. Divide by (x − 1): x³ − 6x² + 11x − 6 = (x − 1)(x² − 5x + 6)
3. Factor the quadratic: (x − 1)(x − 2)(x − 3) = 0
4. Answer: x = 1, 2, 3

For higher-degree equations without nice whole-number roots, an equation solver uses numeric methods and returns decimal answers. That is expected.

Absolute Value Equations

An absolute value equation like |2x − 3| = 7 splits into two equations, because the inside can be positive or negative.

Example: |2x − 3| = 7

• Case 1: 2x − 3 = 7, so 2x = 10, x = 5
• Case 2: 2x − 3 = −7, so 2x = −4, x = −2

Check: |2(5) − 3| = 7 and |2(−2) − 3| = |−7| = 7. Both work.

If the absolute value equals a negative number, like |x + 1| = −4, there is no solution, because absolute values are never negative.

Radical Equations And Extraneous Solutions

A radical equation has the variable under a square root. Solve it by isolating the root and squaring both sides, then always check every answer. Squaring can create extraneous solutions, which are values that come out of the algebra but do not work in the original equation.

Example: √(x + 7) = x − 5

1. Square both sides: x + 7 = x² − 10x + 25
2. Set to zero: x² − 11x + 18 = 0
3. Factor: (x − 9)(x − 2) = 0, so x = 9 or x = 2
4. Check x = 9: √16 = 4 and 9 − 5 = 4. Works.
5. Check x = 2: √9 = 3 but 2 − 5 = −3. Fails, so x = 2 is extraneous.

Answer: x = 9 only.

A square root is never negative, so any solution that makes the right side negative must be thrown out.

Rational Equations

A rational equation has the variable in a denominator. Multiply both sides by the common denominator to clear the fractions, solve, then reject any value that makes a denominator zero.

Example: x/(x − 3) = 3/(x − 3) + 2

1. Note the restriction: x cannot be 3, because x − 3 would be 0.
2. Multiply every term by (x − 3): x = 3 + 2(x − 3)
3. Simplify: x = 3 + 2x − 6, so x = 2x − 3
4. Solve: x = 3
5. But x = 3 is not allowed. There is no solution.

This is a common trap. A careless solve gives x = 3, which would divide by zero in the original equation.

Exponential Equations

An exponential equation has the variable in an exponent, like 5^x = 20. If both sides can be written with the same base, set the exponents equal. If not, take a logarithm of both sides.

Same Base

Example: 3^(2x − 1) = 27

1. Write 27 as a power of 3: 27 = 3³
2. Set the exponents equal: 2x − 1 = 3
3. Solve: x = 2

Different Bases

Example: 5^x = 20

1. Take the natural log of both sides: ln(5^x) = ln(20)
2. Bring the exponent down: x · ln(5) = ln(20)
3. Divide: x = ln(20) ÷ ln(5) ≈ 1.861

Check: 5^1.861 ≈ 20.

Logarithmic Equations

A logarithmic equation has the variable inside a log. Rewrite it in exponential form, solve, and reject any answer that would make a log's input zero or negative.

Rewriting In Exponential Form

Example: log₂(x + 3) = 5

1. Rewrite: x + 3 = 2⁵ = 32
2. Solve: x = 29

Combining Logs First

Example: log(x) + log(x − 3) = 1 (base 10)

1. Combine using the product rule: log(x(x − 3)) = 1
2. Rewrite: x(x − 3) = 10¹ = 10
3. Set to zero: x² − 3x − 10 = 0, so (x − 5)(x + 2) = 0
4. x = 5 or x = −2
5. Reject x = −2, because log(−2) is undefined. Answer: x = 5.

Systems Of Equations

A system is two or more equations that share variables. The solution is the set of values that makes every equation true at the same time.

Three Ways To Solve A System

Example (substitution): 2x + 3y = 12 and x − y = 1

1. From the second equation: x = y + 1
2. Substitute: 2(y + 1) + 3y = 12, so 5y + 2 = 12
3. Solve: y = 2, then x = 3
4. Answer: x = 3, y = 2

Systems With Three Variables

With three variables, use elimination to reduce the system to two equations, then to one.

Example: x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2

1. Add equations 1 and 3 to cancel z: 2x + 3y = 8
2. Add equations 2 and 3 to cancel z: 3x + y = 5
3. Solve these two: from 3x + y = 5, y = 5 − 3x. Then 2x + 3(5 − 3x) = 8, so −7x = −7 and x = 1
4. Back-substitute: y = 2, then z = 6 − 1 − 2 = 3
5. Answer: x = 1, y = 2, z = 3

Inequalities

An inequality uses <, >, ≤ or ≥ instead of =. You solve it like an equation, with one big exception: flip the sign when you multiply or divide by a negative number. As Math is Fun's guide to solving inequalities puts it, when multiplying or dividing by a negative number, reverse the inequality.

Example: −2x + 5 > 11

1. Subtract 5: −2x > 6
2. Divide by −2 and flip the sign: x < −3
3. Answer: x < −3, which is (−∞, −3) in interval notation

Check: try x = −4. −2(−4) + 5 = 13, and 13 > 11 is true.

How To Check An Equation Solver's Answer

The fastest way to check any solution is to plug it back into the original equation, not a later step. Both sides should give the same number.

• Use the original equation. Extraneous solutions often pass a check on a later line but fail the original.
• Check every solution. A quadratic or radical equation can have one good answer and one bad one.
• Watch the domain. Reject values that divide by zero, take a square root of a negative (in real numbers) or take a log of zero or a negative.
• Estimate. If x = 6 and the left side is x/3 + 5, you expect 7. A wildly different value means a typo.
• Graph both sides. The solutions are where the two graphs meet.

Common Equation Solving Mistakes

Most wrong answers come from a few repeat errors. Here is how to spot and fix them.

The last one is easy to miss. Dividing both sides by x throws away the solution x = 0.

Summary

An equation solver finds the values that make both sides equal, but the method changes with the type of equation. Key takeaways:

• Linear: isolate the variable; clear fractions first.
• Quadratic: factor, use square roots or use the quadratic formula; the discriminant predicts the number of real solutions.
• Absolute value: split into two cases.
• Radical and rational: always check for extraneous solutions and denominator restrictions.
• Exponential and logarithmic: match bases or use logs; reject log inputs that are zero or negative.
• Systems: use substitution or elimination; three-variable systems reduce step by step.
• Inequalities: flip the sign when multiplying or dividing by a negative.
• Always check in the original equation.

Next step: Type one of the examples above into the equation solver at the top of this page, compare its steps with the ones here, then try an equation from your own homework. For more free tools and guides, visit the Calc Solver home page.

Frequently Asked Questions