Equation Solver

Q: How are cubic equations solved?

The solver tests candidate rational roots p/q, divides out each root it finds with synthetic division, and solves what is left. Remaining irrational roots are found numerically.

Q: How many solutions can a polynomial equation have?

A polynomial equation of degree n has at most n real solutions, and exactly n complex solutions when repeated roots are counted. The quartic x^4 - 5x^2 + 4 = 0 has four real roots: -2, -1, 1 and 2.

Q: How does the rational root theorem help?

It limits the possible rational roots to fractions p/q where p divides the constant term and q divides the leading coefficient. That gives a short list of candidates to test instead of guessing.

About the Equation Solver

An equation solver has to recognize what kind of problem it is facing. A linear equation needs inverse operations, a quadratic needs the discriminant, and a cubic or quartic can be broken down with the rational root theorem.

This solver collects terms into a polynomial when it can and solves it exactly. For equations such as $2^x = 32$ or $\sin x = 0.5$ it locates real roots numerically and lists them in a stated interval. Without an equals sign it simplifies or evaluates the expression instead.

What is an equation solver?

An equation solver takes an equation in one unknown and returns every value that satisfies it, together with the method. The method depends on the kind of equation. A linear equation such as $5x - 3 = 12$ needs only inverse operations. A polynomial equation such as $x^3 - 6x^2 + 11x - 6 = 0$ needs its roots found one by one. An equation like $2^x = 32$ has the unknown in an exponent and is handled differently again.

This solver first rewrites the equation as one expression equal to zero, then checks whether that expression is a polynomial. If it is, the roots are found exactly; if not, real roots are located numerically on a stated interval. If you enter an expression with no equals sign, it is expanded and simplified, so $(x + 3)^2$ becomes $x^2 + 6x + 9$ .

How to use the Equation Solver

Enter an equation such as x^3 − 6x^2 + 11x − 6 = 0, or an expression such as (x + 3)^2.
Use ^ for powers, * for multiplication (optional between a number and a letter) and sqrt( ) for roots.
Press Solve. The interpretation line tells you which method was chosen.

Formulas

Rational root theorem

a_n x^n + \cdots + a_0 = 0,\ x = \frac{p}{q} \Rightarrow p \mid a_0,\ q \mid a_n

Quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Factor theorem

P(r) = 0 \iff (x - r) \text{ is a factor of } P(x)

Worked example: x³ − 6x² + 11x − 6 = 0

The worked example solves the cubic $x^3 - 6x^2 + 11x - 6 = 0$ . By the rational root theorem any rational root must divide 6, so the solver tests the candidates $\pm1, \pm2, \pm3, \pm6$ . It finds $x = 1$ , divides it out to leave $x^2 - 5x + 6$ , then finds $x = 2$ , leaving $x - 3$ , and finally $x = 3$ . Substituting each root into the original cubic gives 0, which the checks confirm.

The solution below is generated by the same engine that powers the calculator above.

Final answer

x = 1,\ x = 2,\ x = 3

How we read your problem

A polynomial equation of degree 3 in x.

{x}^{3}-6{x}^{2}+11x-6 = 0

Step-by-step solution

Start with the equation
${x}^{3}-6{x}^{2}+11x-6 = 0$
Move every term to one side
$x^{3} - 6x^{2} + 11x - 6 = 0$
Test candidates from the rational root theorem: x = 1 is a root
Synthetic division by (x - 1) leaves a polynomial of degree 2.
$x^{2} - 5x + 6 = 0$
Test candidates from the rational root theorem: x = 2 is a root
Synthetic division by (x - 2) leaves a polynomial of degree 1.
$x - 3 = 0$
Test candidates from the rational root theorem: x = 3 is a root
Synthetic division by (x - 3) leaves a polynomial of degree 0.
Collect the real solutions
$x = 1,\quad x = 2,\quad x = 3$

Formulas used

Rational root theorem
$x = \frac{p}{q},\ p \mid a_0,\ q \mid a_n$

A degree-3 polynomial has exactly 3 roots counted with multiplicity (some may be complex). Exact rational roots are found first, then the rest are solved exactly or approximated.

Verification

x = 1: left side = 0, right side = 0 ✓
x = 2: left side = 0, right side = 0 ✓
x = 3: left side = 0, right side = 0 ✓

Detailed explanation

Polynomial equations, step by step

The factor theorem says that $r$ is a root of a polynomial exactly when $x - r$ is a factor of it. So every root the solver finds lets it divide the polynomial by $x - r$ with synthetic division, lowering the degree by one. Once a quadratic remains, the discriminant and the quadratic formula finish the job exactly. This is how a quartic such as $x^4 - 5x^2 + 4 = 0$ resolves into the four roots $-2, -1, 1$ and $2$ .

If the remaining factor has no rational roots, its real roots are approximated numerically and marked with ≈, while complex roots are reported for quadratic pieces.

Non-polynomial equations

For exponential, logarithmic and trigonometric equations the solver writes the equation as $f(x) = 0$ , scans the interval from −100 to 100 for places where $f$ changes sign, and narrows each one down by bisection. For $2^x = 32$ it finds $x \approx 5$ and substitutes it back to show both sides equal 32. Because only that interval is searched, the solver says so in the steps; periodic equations such as $\sin x = 0.5$ have infinitely many solutions, and only those in the interval are listed.

Common mistakes

Writing 2x^2 when you mean (2x)^2; powers bind tighter than multiplication.
Expecting a single answer from a cubic; a degree- $n$ polynomial can have up to $n$ real roots.
Typing sin x without brackets; write sin(x).
Dividing both sides by a factor that contains $x$ and losing the roots where that factor is zero.
Forgetting to move every term to one side before trying to factor.

Tips

Move everything to one side and factor out common terms before reaching for formulas.
Use the graphing calculator to see how many real roots to expect.
For exponential equations, check whether both sides can be written with the same base.
If all the coefficients add up to 0, then $x = 1$ is a root; that is the case in the worked example.
Read the interpretation line first; it tells you how the input was understood and which method was used.
Factor out a common factor such as $x$ before dividing; $x^3 = 4x$ has three solutions, not two.

Frequently asked questions

What types of equations can this solver handle?

Linear, quadratic and polynomial equations of any degree, rational equations with x in a denominator, and single-variable equations with exponentials, logarithms or trig functions (solved numerically).

How are cubic equations solved?

The solver tests candidate rational roots $\frac{p}{q}$ , divides out each root it finds with synthetic division, and solves what is left. Remaining irrational roots are found numerically.

Why does the answer say ≈?

An approximate sign means the root was located numerically, to about 10 significant digits, because no exact closed form was used.

Can I simplify an expression without an equals sign?

Yes. Enter an expression such as (x + 3)^2 to expand and simplify it, or a numeric expression to evaluate it.

Does it find complex roots?

Complex roots are given for quadratics and for polynomials after real roots are factored out. Non-polynomial equations are solved over the real numbers.

Are the answers checked?

Yes. Each root is substituted into the original equation and the residual is shown in the verification section.

How many solutions can a polynomial equation have?

A polynomial equation of degree $n$ has at most $n$ real solutions, and exactly $n$ complex solutions when repeated roots are counted. The quartic $x^4 - 5x^2 + 4 = 0$ has four real roots: $-2, -1, 1$ and $2$ .

How does the rational root theorem help?

It limits the possible rational roots to fractions $\frac{p}{q}$ where $p$ divides the constant term and $q$ divides the leading coefficient. That gives a short list of candidates to test instead of guessing.

Can I simplify an expression without solving it?

Yes. Leave out the equals sign and the calculator expands and combines like terms instead of solving. Entering $(x + 3)^2$ returns $x^2 + 6x + 9$ .

Solve any equation

Enter an equation or an expression above. The solver detects its type, chooses the method and checks every root it reports.

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