Solve 3sqrt(x ^ 2 + 2x + 2) = 7 + x | 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

$$3\sqrt{x^{2}+2x+2}=7+x$$

Solve for x

$x = \frac{3 \sqrt{7} - 1}{4} \approx 1.734313483$
$x=\frac{-3\sqrt{7}-1}{4}\approx -2.234313483$

Solution Steps

Square both sides of the equation.

$$\left(3\sqrt{x^{2}+2x+2}\right)^{2}=\left(7+x\right)^{2}$$

Expand $\left(3\sqrt{x^{2}+2x+2}\right)^{2}$.

$$3^{2}\left(\sqrt{x^{2}+2x+2}\right)^{2}=\left(7+x\right)^{2}$$

Calculate $3$ to the power of $2$ and get $9$.

$$9\left(\sqrt{x^{2}+2x+2}\right)^{2}=\left(7+x\right)^{2}$$

Calculate $\sqrt{x^{2}+2x+2}$ to the power of $2$ and get $x^{2}+2x+2$.

$$9\left(x^{2}+2x+2\right)=\left(7+x\right)^{2}$$

Use the distributive property to multiply $9$ by $x^{2}+2x+2$.

$$9x^{2}+18x+18=\left(7+x\right)^{2}$$

Use binomial theorem $\left(a+b\right)^{2}=a^{2}+2ab+b^{2}$ to expand $\left(7+x\right)^{2}$.

$$9x^{2}+18x+18=49+14x+x^{2}$$

Subtract $49$ from both sides.

$$9x^{2}+18x+18-49=14x+x^{2}$$

Subtract $49$ from $18$ to get $-31$.

$$9x^{2}+18x-31=14x+x^{2}$$

Subtract $14x$ from both sides.

$$9x^{2}+18x-31-14x=x^{2}$$

Combine $18x$ and $-14x$ to get $4x$.

$$9x^{2}+4x-31=x^{2}$$

Subtract $x^{2}$ from both sides.

$$9x^{2}+4x-31-x^{2}=0$$

Combine $9x^{2}$ and $-x^{2}$ to get $8x^{2}$.

$$8x^{2}+4x-31=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $8$ for $a$, $4$ for $b$, and $-31$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-4±\sqrt{4^{2}-4\times 8\left(-31\right)}}{2\times 8}$$

Square $4$.

$$x=\frac{-4±\sqrt{16-4\times 8\left(-31\right)}}{2\times 8}$$

Multiply $-4$ times $8$.

$$x=\frac{-4±\sqrt{16-32\left(-31\right)}}{2\times 8}$$

Multiply $-32$ times $-31$.

$$x=\frac{-4±\sqrt{16+992}}{2\times 8}$$

Add $16$ to $992$.

$$x=\frac{-4±\sqrt{1008}}{2\times 8}$$

Take the square root of $1008$.

$$x=\frac{-4±12\sqrt{7}}{2\times 8}$$

Multiply $2$ times $8$.

$$x=\frac{-4±12\sqrt{7}}{16}$$

Now solve the equation $x=\frac{-4±12\sqrt{7}}{16}$ when $±$ is plus. Add $-4$ to $12\sqrt{7}$.

$$x=\frac{12\sqrt{7}-4}{16}$$

Divide $-4+12\sqrt{7}$ by $16$.

$$x=\frac{3\sqrt{7}-1}{4}$$

Now solve the equation $x=\frac{-4±12\sqrt{7}}{16}$ when $±$ is minus. Subtract $12\sqrt{7}$ from $-4$.

$$x=\frac{-12\sqrt{7}-4}{16}$$

Divide $-4-12\sqrt{7}$ by $16$.

$$x=\frac{-3\sqrt{7}-1}{4}$$

The equation is now solved.

$$x=\frac{3\sqrt{7}-1}{4}$$ $$x=\frac{-3\sqrt{7}-1}{4}$$

Substitute $\frac{3\sqrt{7}-1}{4}$ for $x$ in the equation $3\sqrt{x^{2}+2x+2}=7+x$.

$$3\sqrt{\left(\frac{3\sqrt{7}-1}{4}\right)^{2}+2\times \frac{3\sqrt{7}-1}{4}+2}=7+\frac{3\sqrt{7}-1}{4}$$

Simplify. The value $x=\frac{3\sqrt{7}-1}{4}$ satisfies the equation.

$$\frac{27}{4}+\frac{3}{4}\times 7^{\frac{1}{2}}=\frac{27}{4}+\frac{3}{4}\times 7^{\frac{1}{2}}$$

Substitute $\frac{-3\sqrt{7}-1}{4}$ for $x$ in the equation $3\sqrt{x^{2}+2x+2}=7+x$.

$$3\sqrt{\left(\frac{-3\sqrt{7}-1}{4}\right)^{2}+2\times \frac{-3\sqrt{7}-1}{4}+2}=7+\frac{-3\sqrt{7}-1}{4}$$

Simplify. The value $x=\frac{-3\sqrt{7}-1}{4}$ satisfies the equation.

$$\frac{27}{4}-\frac{3}{4}\times 7^{\frac{1}{2}}=\frac{27}{4}-\frac{3}{4}\times 7^{\frac{1}{2}}$$

List all solutions of $3\sqrt{x^{2}+2x+2}=x+7$.

$$x=\frac{3\sqrt{7}-1}{4}$$ $$x=\frac{-3\sqrt{7}-1}{4}$$