Solve by Completing the Square 2y^2-3y-6=0

$2 y^{2} - 3 y - 6 = 0$

Step 1

Add $6$ to both sides of the equation.

$2 y^{2} - 3 y = 6$

Step 2

Divide each term in $2 y^{2} - 3 y = 6$ by $2$ and simplify.

Divide each term in $2 y^{2} - 3 y = 6$ by $2$ .

$\frac{2 y^{2}}{2} + \frac{- 3 y}{2} = \frac{6}{2}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 y^{2}}{2} + \frac{- 3 y}{2} = \frac{6}{2}$

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

$y^{2} + \frac{- 3 y}{2} = \frac{6}{2}$

Move the negative in front of the fraction.

$y^{2} - \frac{3 y}{2} = \frac{6}{2}$

Simplify the right side.

Divide $6$ by $2$ .

$y^{2} - \frac{3 y}{2} = 3$

Step 3

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 4

Add the term to each side of the equation.

$y^{2} - \frac{3 y}{2} + {(- \frac{3}{4})}^{2} = 3 + {(- \frac{3}{4})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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

$y^{2} - \frac{3 y}{2} + {(- 1)}^{2} {(\frac{3}{4})}^{2} = 3 + {(- \frac{3}{4})}^{2}$

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

$y^{2} - \frac{3 y}{2} + {(- 1)}^{2} \frac{3^{2}}{4^{2}} = 3 + {(- \frac{3}{4})}^{2}$

Raise $- 1$ to the power of $2$ .

$y^{2} - \frac{3 y}{2} + 1 \frac{3^{2}}{4^{2}} = 3 + {(- \frac{3}{4})}^{2}$

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

$y^{2} - \frac{3 y}{2} + \frac{3^{2}}{4^{2}} = 3 + {(- \frac{3}{4})}^{2}$

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

$y^{2} - \frac{3 y}{2} + \frac{9}{4^{2}} = 3 + {(- \frac{3}{4})}^{2}$

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

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = 3 + {(- \frac{3}{4})}^{2}$

Simplify the right side.

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

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = 3 + {(- 1)}^{2} {(\frac{3}{4})}^{2}$

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

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = 3 + {(- 1)}^{2} \frac{3^{2}}{4^{2}}$

Raise $- 1$ to the power of $2$ .

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = 3 + 1 \frac{3^{2}}{4^{2}}$

Multiply $\frac{3^{2}}{4^{2}}$ by $1$ .

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = 3 + \frac{3^{2}}{4^{2}}$

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

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = 3 + \frac{9}{4^{2}}$

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

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = 3 + \frac{9}{16}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = 3 \cdot \frac{16}{16} + \frac{9}{16}$

Combine $3$ and $\frac{16}{16}$ .

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = \frac{3 \cdot 16}{16} + \frac{9}{16}$

Combine the numerators over the common denominator.

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = \frac{3 \cdot 16 + 9}{16}$

Simplify the numerator.

Multiply $3$ by $16$ .

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = \frac{48 + 9}{16}$

Add $48$ and $9$ .

$y^{2} - \frac{3 y}{2} + \frac{9}{16} = \frac{57}{16}$

Step 6

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

${(y - \frac{3}{4})}^{2} = \frac{57}{16}$

Step 7

Solve the equation for $y$ .

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

$y - \frac{3}{4} = \pm \sqrt{\frac{57}{16}}$

Simplify $\pm \sqrt{\frac{57}{16}}$ .

Rewrite $\sqrt{\frac{57}{16}}$ as $\frac{\sqrt{57}}{\sqrt{16}}$ .

$y - \frac{3}{4} = \pm \frac{\sqrt{57}}{\sqrt{16}}$

Simplify the denominator.

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

$y - \frac{3}{4} = \pm \frac{\sqrt{57}}{\sqrt{4^{2}}}$

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

$y - \frac{3}{4} = \pm \frac{\sqrt{57}}{4}$

Add $\frac{3}{4}$ to both sides of the equation.

$y = \pm \frac{\sqrt{57}}{4} + \frac{3}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = \pm \frac{\sqrt{57}}{4} + \frac{3}{4}$

Decimal Form:

$y = 2.63745860 \dots, - 1.13745860 \dots$

Solve by Completing the Square 2y^2-3y-6=0

Related Questions

Support LCMGCF.com by disabling your adblocker.