Solve 2x ^ 3 - 3x ^ 2 - 11x + 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

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

Solve for x

$x=\frac{1}{2}=0.5$
$x=3$
$x=-2$

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 $2$. List all candidates $\frac{p}{q}$.

$$±3,±6,±\frac{3}{2},±1,±2,±\frac{1}{2}$$

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.

$$x=-2$$

By Factor theorem, $x-k$ is a factor of the polynomial for each root $k$. Divide $2x^{3}-3x^{2}-11x+6$ by $x+2$ to get $2x^{2}-7x+3$. Solve the equation where the result equals to $0$.

$$2x^{2}-7x+3=0$$

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 $2$ for $a$, $-7$ for $b$, and $3$ for $c$ in the quadratic formula.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 2\times 3}}{2\times 2}$$

Do the calculations.

$$x=\frac{7±5}{4}$$

Solve the equation $2x^{2}-7x+3=0$ when $±$ is plus and when $±$ is minus.

$$x=\frac{1}{2}$$ $$x=3$$

List all found solutions.

$$x=-2$$ $$x=\frac{1}{2}$$ $$x=3$$