Solve x ^ 3 + 2x ^ 2 - 5x - 6 = 0 | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$x^{3}+2x^{2}-5x-6=0$$

Solve for x

$x=-1$
$x=2$
$x=-3$

Solution Steps

By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-6$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.

$$±6,±3,±2,±1$$

Find one such root by trying out all the integer values, starting from the smallest by absolute value. If no integer roots are found, try out fractions.

$$x=-1$$

By Factor theorem, $x-k$ is a factor of the polynomial for each root $k$. Divide $x^{3}+2x^{2}-5x-6$ by $x+1$ to get $x^{2}+x-6$. Solve the equation where the result equals to $0$.

$$x^{2}+x-6=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. Substitute $1$ for $a$, $1$ for $b$, and $-6$ for $c$ in the quadratic formula.

$$x=\frac{-1±\sqrt{1^{2}-4\times 1\left(-6\right)}}{2}$$

Do the calculations.

$$x=\frac{-1±5}{2}$$

Solve the equation $x^{2}+x-6=0$ when $±$ is plus and when $±$ is minus.

$$x=-3$$ $$x=2$$

List all found solutions.

$$x=-1$$ $$x=-3$$ $$x=2$$