Solve by Completing the Square 2x^2+3x=20

$2 x^{2} + 3 x = 20$

Step 1

Divide each term in $2 x^{2} + 3 x = 20$ by $2$ and simplify.

Divide each term in $2 x^{2} + 3 x = 20$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{3 x}{2} = \frac{20}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{3 x}{2} = \frac{20}{2}$

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

$x^{2} + \frac{3 x}{2} = \frac{20}{2}$

Simplify the right side.

Divide $20$ by $2$ .

$x^{2} + \frac{3 x}{2} = 10$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{3}{4})}^{2}$

Step 3

Add the term to each side of the equation.

$x^{2} + \frac{3 x}{2} + {(\frac{3}{4})}^{2} = 10 + {(\frac{3}{4})}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{3}{4}$ .

$x^{2} + \frac{3 x}{2} + \frac{3^{2}}{4^{2}} = 10 + {(\frac{3}{4})}^{2}$

Raise $3$ to the power of $2$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{4^{2}} = 10 + {(\frac{3}{4})}^{2}$

Raise $4$ to the power of $2$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 10 + {(\frac{3}{4})}^{2}$

Simplify the right side.

Simplify $10 + {(\frac{3}{4})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{3}{4}$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 10 + \frac{3^{2}}{4^{2}}$

Raise $3$ to the power of $2$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 10 + \frac{9}{4^{2}}$

Raise $4$ to the power of $2$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 10 + \frac{9}{16}$

To write $10$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = 10 \cdot \frac{16}{16} + \frac{9}{16}$

Combine $10$ and $\frac{16}{16}$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = \frac{10 \cdot 16}{16} + \frac{9}{16}$

Combine the numerators over the common denominator.

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = \frac{10 \cdot 16 + 9}{16}$

Simplify the numerator.

Multiply $10$ by $16$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = \frac{160 + 9}{16}$

Add $160$ and $9$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16} = \frac{169}{16}$

Step 5

Factor the perfect trinomial square into ${(x + \frac{3}{4})}^{2}$ .

${(x + \frac{3}{4})}^{2} = \frac{169}{16}$

Step 6

Solve the equation for $x$ .

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

$x + \frac{3}{4} = \pm \sqrt{\frac{169}{16}}$

Simplify $\pm \sqrt{\frac{169}{16}}$ .

Rewrite $\sqrt{\frac{169}{16}}$ as $\frac{\sqrt{169}}{\sqrt{16}}$ .

$x + \frac{3}{4} = \pm \frac{\sqrt{169}}{\sqrt{16}}$

Simplify the numerator.

Rewrite $169$ as $13^{2}$ .

$x + \frac{3}{4} = \pm \frac{\sqrt{13^{2}}}{\sqrt{16}}$

Pull terms out from under the radical, assuming positive real numbers.

$x + \frac{3}{4} = \pm \frac{13}{\sqrt{16}}$

Simplify the denominator.

Rewrite $16$ as $4^{2}$ .

$x + \frac{3}{4} = \pm \frac{13}{\sqrt{4^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$x + \frac{3}{4} = \pm \frac{13}{4}$

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.

$x + \frac{3}{4} = \frac{13}{4}$

Move all terms not containing $x$ to the right side of the equation.

Subtract $\frac{3}{4}$ from both sides of the equation.

$x = \frac{13}{4} - \frac{3}{4}$

Combine the numerators over the common denominator.

$x = \frac{13 - 3}{4}$

Subtract $3$ from $13$ .

$x = \frac{10}{4}$

Cancel the common factor of $10$ and $4$ .

Factor $2$ out of $10$ .

$x = \frac{2 (5)}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 5}{2 \cdot 2}$

Cancel the common factor.

$x = \frac{2 \cdot 5}{2 \cdot 2}$

Rewrite the expression.

$x = \frac{5}{2}$

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

$x + \frac{3}{4} = - \frac{13}{4}$

Move all terms not containing $x$ to the right side of the equation.

Subtract $\frac{3}{4}$ from both sides of the equation.

$x = - \frac{13}{4} - \frac{3}{4}$

Combine the numerators over the common denominator.

$x = \frac{- 13 - 3}{4}$

Subtract $3$ from $- 13$ .

$x = \frac{- 16}{4}$

Divide $- 16$ by $4$ .

$x = - 4$

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

$x = \frac{5}{2}, - 4$

Solve by Completing the Square 2x^2+3x=20

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