Solve (x+2)=(x+(4)/(11))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

$$(x+2)=(x+ \frac{ 4 }{ 11 } )x$$

Solve for x

$x = \frac{3 \sqrt{113} + 7}{22} \approx 1.767747156$
$x=\frac{7-3\sqrt{113}}{22}\approx -1.13138352$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Use the distributive property to multiply $x+\frac{4}{11}$ by $x$.

$$x+2=x^{2}+\frac{4}{11}x$$

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

$$x+2-x^{2}=\frac{4}{11}x$$

Subtract $\frac{4}{11}x$ from both sides.

$$x+2-x^{2}-\frac{4}{11}x=0$$

Combine $x$ and $-\frac{4}{11}x$ to get $\frac{7}{11}x$.

$$\frac{7}{11}x+2-x^{2}=0$$

All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.

$$-x^{2}+\frac{7}{11}x+2=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-1$ for $a$, $\frac{7}{11}$ for $b$, and $2$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\frac{7}{11}±\sqrt{\left(\frac{7}{11}\right)^{2}-4\left(-1\right)\times 2}}{2\left(-1\right)}$$

Square $\frac{7}{11}$ by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{7}{11}±\sqrt{\frac{49}{121}-4\left(-1\right)\times 2}}{2\left(-1\right)}$$

Multiply $-4$ times $-1$.

$$x=\frac{-\frac{7}{11}±\sqrt{\frac{49}{121}+4\times 2}}{2\left(-1\right)}$$

Multiply $4$ times $2$.

$$x=\frac{-\frac{7}{11}±\sqrt{\frac{49}{121}+8}}{2\left(-1\right)}$$

Add $\frac{49}{121}$ to $8$.

$$x=\frac{-\frac{7}{11}±\sqrt{\frac{1017}{121}}}{2\left(-1\right)}$$

Take the square root of $\frac{1017}{121}$.

$$x=\frac{-\frac{7}{11}±\frac{3\sqrt{113}}{11}}{2\left(-1\right)}$$

Multiply $2$ times $-1$.

$$x=\frac{-\frac{7}{11}±\frac{3\sqrt{113}}{11}}{-2}$$

Now solve the equation $x=\frac{-\frac{7}{11}±\frac{3\sqrt{113}}{11}}{-2}$ when $±$ is plus. Add $-\frac{7}{11}$ to $\frac{3\sqrt{113}}{11}$.

$$x=\frac{3\sqrt{113}-7}{-2\times 11}$$

Divide $\frac{-7+3\sqrt{113}}{11}$ by $-2$.

$$x=\frac{7-3\sqrt{113}}{22}$$

Now solve the equation $x=\frac{-\frac{7}{11}±\frac{3\sqrt{113}}{11}}{-2}$ when $±$ is minus. Subtract $\frac{3\sqrt{113}}{11}$ from $-\frac{7}{11}$.

$$x=\frac{-3\sqrt{113}-7}{-2\times 11}$$

Divide $\frac{-7-3\sqrt{113}}{11}$ by $-2$.

$$x=\frac{3\sqrt{113}+7}{22}$$

The equation is now solved.

$$x=\frac{7-3\sqrt{113}}{22}$$ $$x=\frac{3\sqrt{113}+7}{22}$$

Steps for Completing the Square

Use the distributive property to multiply $x+\frac{4}{11}$ by $x$.

$$x+2=x^{2}+\frac{4}{11}x$$

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

$$x+2-x^{2}=\frac{4}{11}x$$

Subtract $\frac{4}{11}x$ from both sides.

$$x+2-x^{2}-\frac{4}{11}x=0$$

Combine $x$ and $-\frac{4}{11}x$ to get $\frac{7}{11}x$.

$$\frac{7}{11}x+2-x^{2}=0$$

Subtract $2$ from both sides. Anything subtracted from zero gives its negation.

$$\frac{7}{11}x-x^{2}=-2$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form $x^{2}+bx=c$.

$$-x^{2}+\frac{7}{11}x=-2$$

Divide both sides by $-1$.

$$\frac{-x^{2}+\frac{7}{11}x}{-1}=-\frac{2}{-1}$$

Dividing by $-1$ undoes the multiplication by $-1$.

$$x^{2}+\frac{\frac{7}{11}}{-1}x=-\frac{2}{-1}$$

Divide $\frac{7}{11}$ by $-1$.

$$x^{2}-\frac{7}{11}x=-\frac{2}{-1}$$

Divide $-2$ by $-1$.

$$x^{2}-\frac{7}{11}x=2$$

Divide $-\frac{7}{11}$, the coefficient of the $x$ term, by $2$ to get $-\frac{7}{22}$. Then add the square of $-\frac{7}{22}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{11}x+\left(-\frac{7}{22}\right)^{2}=2+\left(-\frac{7}{22}\right)^{2}$$

Square $-\frac{7}{22}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{11}x+\frac{49}{484}=2+\frac{49}{484}$$

Add $2$ to $\frac{49}{484}$.

$$x^{2}-\frac{7}{11}x+\frac{49}{484}=\frac{1017}{484}$$

Factor $x^{2}-\frac{7}{11}x+\frac{49}{484}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{7}{22}\right)^{2}=\frac{1017}{484}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{22}\right)^{2}}=\sqrt{\frac{1017}{484}}$$

Simplify.

$$x-\frac{7}{22}=\frac{3\sqrt{113}}{22}$$ $$x-\frac{7}{22}=-\frac{3\sqrt{113}}{22}$$

Add $\frac{7}{22}$ to both sides of the equation.

$$x=\frac{3\sqrt{113}+7}{22}$$ $$x=\frac{7-3\sqrt{113}}{22}$$