Solve (x ^ 2 + x - 5)(x ^ 2 + x + 1) = - 9 | 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

$$(x^{2}+x-5)(x^{2}+x+1)=-9$$

Solve for x

$x=-2$
$x=1$

Solution Steps

Use the distributive property to multiply $x^{2}+x-5$ by $x^{2}+x+1$ and combine like terms.

$$x^{4}+2x^{3}-3x^{2}-4x-5=-9$$

Add $9$ to both sides.

$$x^{4}+2x^{3}-3x^{2}-4x-5+9=0$$

Add $-5$ and $9$ to get $4$.

$$x^{4}+2x^{3}-3x^{2}-4x+4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $4$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.

$$±4,±2,±1$$

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 $x^{4}+2x^{3}-3x^{2}-4x+4$ by $x-1$ to get $x^{3}+3x^{2}-4$. Solve the equation where the result equals to $0$.

$$x^{3}+3x^{2}-4=0$$

By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-4$ and $q$ divides the leading coefficient $1$. List all candidates $\frac{p}{q}$.

$$±4,±2,±1$$

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 $x^{3}+3x^{2}-4$ by $x-1$ to get $x^{2}+4x+4$. Solve the equation where the result equals to $0$.

$$x^{2}+4x+4=0$$

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 $a$, $4$ for $b$, and $4$ for $c$ in the quadratic formula.

$$x=\frac{-4±\sqrt{4^{2}-4\times 1\times 4}}{2}$$

Do the calculations.

$$x=\frac{-4±0}{2}$$

Solutions are the same.

$$x=-2$$

List all found solutions.

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