Solve for x 3x(2x+3)=2x(x+4.5)+2

$3 x (2 x + 3) = 2 x (x + 4.5) + 2$

Simplify $3 x (2 x + 3)$ .

Rewrite.

$0 + 0 + 3 x (2 x + 3) = 2 x (x + 4.5) + 2$

Simplify by multiplying through.

Apply the distributive property.

$3 x (2 x) + 3 x \cdot 3 = 2 x (x + 4.5) + 2$

Simplify the expression.

Rewrite using the commutative property of multiplication.

$3 \cdot 2 x \cdot x + 3 x \cdot 3 = 2 x (x + 4.5) + 2$

Multiply $3$ by $3$ .

$3 \cdot 2 x \cdot x + 9 x = 2 x (x + 4.5) + 2$

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$3 \cdot 2 (x \cdot x) + 9 x = 2 x (x + 4.5) + 2$

Multiply $x$ by $x$ .

$3 \cdot 2 x^{2} + 9 x = 2 x (x + 4.5) + 2$

Multiply $3$ by $2$ .

$6 x^{2} + 9 x = 2 x (x + 4.5) + 2$

Simplify each term.

Apply the distributive property.

$6 x^{2} + 9 x = 2 x \cdot x + 2 x \cdot 4.5 + 2$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$6 x^{2} + 9 x = 2 (x \cdot x) + 2 x \cdot 4.5 + 2$

Multiply $x$ by $x$ .

$6 x^{2} + 9 x = 2 x^{2} + 2 x \cdot 4.5 + 2$

Multiply $4.5$ by $2$ .

$6 x^{2} + 9 x = 2 x^{2} + 9 x + 2$

Move all terms containing $x$ to the left side of the equation.

Subtract $2 x^{2}$ from both sides of the equation.

$6 x^{2} + 9 x - 2 x^{2} = 9 x + 2$

Subtract $9 x$ from both sides of the equation.

$6 x^{2} + 9 x - 2 x^{2} - 9 x = 2$

Subtract $2 x^{2}$ from $6 x^{2}$ .

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

Subtract $9 x$ from $9 x$ .

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

Multiply $0$ by $x$ .

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

Add $4 x^{2}$ and $0$ .

$4 x^{2} = 2$

Divide each term in $4 x^{2} = 2$ by $4$ and simplify.

Divide each term in $4 x^{2} = 2$ by $4$ .

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

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

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

Divide $x^{2}$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$x^{2} = \frac{2 (1)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x^{2} = \frac{2 \cdot 1}{2 \cdot 2}$

Cancel the common factor.

$x^{2} = \frac{2 \cdot 1}{2 \cdot 2}$

Rewrite the expression.

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

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{2}}$

Simplify $\pm \sqrt{\frac{1}{2}}$ .

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{2}}$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{2}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{2}}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{2}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Decimal Form:

$x = 0.70710678 \dots, - 0.70710678 \dots$

Solve for x 3x(2x+3)=2x(x+4.5)+2

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