Online Equation Solver with Steps

Find the exact roots for linear, quadratic, polynomial, trigonometric, exponential, and rational equations. Get clear, step-by-step algebraic manipulations to learn equation solving, simplify complex fractions, and isolate variables instantly.

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The Ultimate Algebraic Engine

Solving equations is the cornerstone of all advanced mathematics. Our algorithmic engine does not just guess answers; it parses the deep mathematical structure of your equality and applies the precise theorem required to isolate the variable.

  • Linear Equations (ax+b=0ax + b = 0): Instantly isolates variables using basic inverse operations.
  • Quadratic Equations (ax2+bx+c=0ax^2 + bx + c = 0): Computes the discriminant (Δ=b2−4ac\Delta = b^2 - 4ac). If Δ>0\Delta > 0, it finds two real roots. If Δ=0\Delta = 0, it finds one repeated real root. If Δ<0\Delta < 0, it successfully computes the two complex/imaginary roots using i=−1i = \sqrt{-1}. It shows solving via factoring, completing the square, and the quadratic formula.
  • High-Degree Polynomials: Can find the roots of cubic (ax3+bx2+cx+d=0ax^3 + bx^2 + cx + d = 0) and quartic equations using advanced techniques like synthetic division, the Rational Root Theorem, and factoring by grouping.
  • Rational Equations: Equations with variables in the denominator (e.g., 1x−2+3x+2=4x2−4\frac{1}{x-2} + \frac{3}{x+2} = \frac{4}{x^2-4}). The solver finds the Least Common Denominator (LCD), multiplies through to eliminate fractions, solves the resulting polynomial, and crucially checks the final answers against the original domain to eliminate extraneous solutions (division by zero).
  • Exponential & Logarithmic Equations: Uses the property ax=b  ⟹  x=log⁡a(b)a^x = b \implies x = \log_a(b) to bring variables out of the exponent. It also condenses logarithmic expressions using log⁡(a)+log⁡(b)=log⁡(ab)\log(a) + \log(b) = \log(ab) to solve complex logarithmic equalities.
  • Trigonometric Equations: Fully supports solving periodic equalities. If you input cos⁡(x)=22\cos(x) = \frac{\sqrt{2}}{2}, it won't just output 45∘45^\circ. It will provide the full, infinite family of solutions (x=π4+2πnx = \frac{\pi}{4} + 2\pi n and x=7π4+2πnx = \frac{7\pi}{4} + 2\pi n) because trigonometric functions repeat forever.

Real-World Applications

The ability to solve complex equations translates directly to high-level careers in STEM.

  • Physics (Kinematics): Using the equation s(t)=−12gt2+v0t+s0s(t) = -\frac{1}{2}gt^2 + v_0t + s_0, physicists can set the height s(t)s(t) to 00 and use our quadratic solver to determine exactly how many seconds it takes for a projectile to hit the ground.
  • Finance & Economics: The compound interest formula A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt} is an exponential equation. If an investor wants to know exactly how many years (tt) it takes to double their money, they must solve the exponential equation using natural logarithms.
  • Chemical Engineering: Balancing chemical equations and calculating reaction rates often requires solving systems of rational and non-linear equations to determine exact molar concentrations.

Exhaustive Step-by-Step Worked Examples

Example 1: Solving a Complex Rational Equation
Solve: 2x+3−1x−3=4x2−9\frac{2}{x+3} - \frac{1}{x-3} = \frac{4}{x^2-9}

  1. Identify the LCD: Notice that the denominator on the right, x2−9x^2 - 9, is a difference of squares that factors perfectly into (x+3)(x−3)(x+3)(x-3). Therefore, the Least Common Denominator for the entire equation is (x+3)(x−3)(x+3)(x-3).
  2. Multiply Through: Multiply every single term in the equation by (x+3)(x−3)(x+3)(x-3) to eliminate all fractions.
    • 2(x+3)(x−3)x+3−1(x+3)(x−3)x−3=4(x+3)(x−3)(x+3)(x−3)\frac{2(x+3)(x-3)}{x+3} - \frac{1(x+3)(x-3)}{x-3} = \frac{4(x+3)(x-3)}{(x+3)(x-3)}
  3. Cancel Factors: This simplifies brilliantly to: 2(x−3)−1(x+3)=42(x-3) - 1(x+3) = 4.
  4. Distribute: 2x−6−x−3=42x - 6 - x - 3 = 4.
  5. Combine Like Terms: x−9=4x - 9 = 4.
  6. Isolate x: Add 9 to both sides to get x=13x = 13.
  7. Check for Extraneous Roots: Plug x=13x=13 back into the original equation denominators. Neither 13+313+3 nor 13−313-3 equals zero, so the solution is valid. Final Answer: x=13x = 13.

Example 2: Solving an Exponential Equation with Logarithms
Solve: 52x−1=1255^{2x-1} = 125

  1. Check for Common Bases: Look at the number 125125. Recognize that 125125 is a perfect power of 55 (specifically, 535^3).
  2. Rewrite the Equation: Substitute 535^3 into the right side: 52x−1=535^{2x-1} = 5^3.
  3. Equate the Exponents: Because the bases are identical (55), the exponents must be absolutely equal for the equation to hold true. Therefore, 2x−1=32x - 1 = 3.
  4. Solve the Linear Equation: Add 1 to both sides: 2x=42x = 4. Divide by 2: x=2x = 2.
  5. Final Answer: x=2x = 2.

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What is an extraneous solution?

An extraneous solution is a "fake" answer that emerges during the algebraic solving process, but does not actually make the original equation true. This most commonly happens when you square both sides of an equation (which can introduce false positives) or when multiplying by a variable expression (which might inadvertently cause division by zero). Our engine automatically tests all answers and discards extraneous solutions.

Can the calculator solve for variables other than x?

Yes. While x is the standard default variable, the engine is completely variable-agnostic. You can type equations using y, z, t, theta, or any other standard letter, and the engine will automatically detect it and isolate that specific variable.

Does it support systems of equations?

Currently, this specific calculator is designed to isolate and solve single-variable equations. Support for solving multi-variable systems of equations (using substitution or elimination matrices) will be added in a future update.

How do I enter a quadratic equation?

Simply type your equation using standard mathematical notation, such as 2x^2 - 5x + 3 = 0. The calculator will automatically detect that the highest power is 2, classify it as a quadratic equation, and apply the appropriate solving method (factoring or the quadratic formula).

What does it mean if the discriminant is negative?

When using the quadratic formula, the discriminant is the value under the square root (b2−4acb^2 - 4ac). If this number is negative, it is mathematically impossible to take its square root using real numbers. In this case, there are no real solutions, and the calculator will output two complex (imaginary) roots using ii.

Can the solver handle fractions and rational equations?

Yes! If you enter an equation with variables in the denominator (like 1x+12=1\frac{1}{x} + \frac{1}{2} = 1), the engine will automatically find the Least Common Denominator (LCD), multiply the entire equation by it to eliminate all fractions, and then solve for the isolated variable.

How does the calculator solve complex logarithmic equations?

It uses the core properties of logarithms. If you have multiple log terms (like log⁡(x)+log⁡(x−1)\log(x) + \log(x-1)), it uses the product rule to condense them into a single logarithm (log⁡(x(x−1))\log(x(x-1))). It then converts the logarithmic equation into an exponential equation to free the variable from the log argument.

Why do trigonometric equations have infinitely many solutions?

Trigonometric functions like sine and cosine are periodic, meaning their wave patterns repeat forever along the xx-axis. Therefore, an equation like sin⁡(x)=0.5\sin(x) = 0.5 is true at infinite regular intervals. The calculator appends terms like +2πn+ 2\pi n to represent this infinite family of solutions.

Can I solve literal equations with multiple unknown letters?

Yes! If you enter an equation with multiple variables, such as the physics formula E=m∗c2E = m*c^2, and specify that you want to solve for mm, the calculator will treat EE and cc as constants and successfully isolate the target variable, resulting in m=Ec2m = \frac{E}{c^2}.

What is the Rational Root Theorem?

It is an advanced algebraic theorem that our calculator uses behind the scenes to solve high-degree polynomials (like cubic or quartic equations). It generates a list of potential rational roots based on the factors of the constant term and the leading coefficient, then uses synthetic division to test them and break the polynomial down.