Solve by Completing the Square 6m^2+5m-3=0

$6 m^{2} + 5 m - 3 = 0$

Step 1

Add $3$ to both sides of the equation.

$6 m^{2} + 5 m = 3$

Step 2

Divide each term in $6 m^{2} + 5 m = 3$ by $6$ and simplify.

Divide each term in $6 m^{2} + 5 m = 3$ by $6$ .

$\frac{6 m^{2}}{6} + \frac{5 m}{6} = \frac{3}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 m^{2}}{6} + \frac{5 m}{6} = \frac{3}{6}$

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

$m^{2} + \frac{5 m}{6} = \frac{3}{6}$

Simplify the right side.

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3$ .

$m^{2} + \frac{5 m}{6} = \frac{3 (1)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

$m^{2} + \frac{5 m}{6} = \frac{3 \cdot 1}{3 \cdot 2}$

Cancel the common factor.

$m^{2} + \frac{5 m}{6} = \frac{3 \cdot 1}{3 \cdot 2}$

Rewrite the expression.

$m^{2} + \frac{5 m}{6} = \frac{1}{2}$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{5}{12})}^{2}$

Step 4

Add the term to each side of the equation.

$m^{2} + \frac{5 m}{6} + {(\frac{5}{12})}^{2} = \frac{1}{2} + {(\frac{5}{12})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{5}{12}$ .

$m^{2} + \frac{5 m}{6} + \frac{5^{2}}{12^{2}} = \frac{1}{2} + {(\frac{5}{12})}^{2}$

Raise $5$ to the power of $2$ .

$m^{2} + \frac{5 m}{6} + \frac{25}{12^{2}} = \frac{1}{2} + {(\frac{5}{12})}^{2}$

Raise $12$ to the power of $2$ .

$m^{2} + \frac{5 m}{6} + \frac{25}{144} = \frac{1}{2} + {(\frac{5}{12})}^{2}$

Simplify the right side.

Simplify $\frac{1}{2} + {(\frac{5}{12})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{5}{12}$ .

$m^{2} + \frac{5 m}{6} + \frac{25}{144} = \frac{1}{2} + \frac{5^{2}}{12^{2}}$

Raise $5$ to the power of $2$ .

$m^{2} + \frac{5 m}{6} + \frac{25}{144} = \frac{1}{2} + \frac{25}{12^{2}}$

Raise $12$ to the power of $2$ .

$m^{2} + \frac{5 m}{6} + \frac{25}{144} = \frac{1}{2} + \frac{25}{144}$

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{72}{72}$ .

$m^{2} + \frac{5 m}{6} + \frac{25}{144} = \frac{1}{2} \cdot \frac{72}{72} + \frac{25}{144}$

Write each expression with a common denominator of $144$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{2}$ by $\frac{72}{72}$ .

$m^{2} + \frac{5 m}{6} + \frac{25}{144} = \frac{72}{2 \cdot 72} + \frac{25}{144}$

Multiply $2$ by $72$ .

$m^{2} + \frac{5 m}{6} + \frac{25}{144} = \frac{72}{144} + \frac{25}{144}$

Combine the numerators over the common denominator.

$m^{2} + \frac{5 m}{6} + \frac{25}{144} = \frac{72 + 25}{144}$

Add $72$ and $25$ .

$m^{2} + \frac{5 m}{6} + \frac{25}{144} = \frac{97}{144}$

Step 6

Factor the perfect trinomial square into ${(m + \frac{5}{12})}^{2}$ .

${(m + \frac{5}{12})}^{2} = \frac{97}{144}$

Step 7

Solve the equation for $m$ .

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

$m + \frac{5}{12} = \pm \sqrt{\frac{97}{144}}$

Simplify $\pm \sqrt{\frac{97}{144}}$ .

Rewrite $\sqrt{\frac{97}{144}}$ as $\frac{\sqrt{97}}{\sqrt{144}}$ .

$m + \frac{5}{12} = \pm \frac{\sqrt{97}}{\sqrt{144}}$

Simplify the denominator.

Rewrite $144$ as $12^{2}$ .

$m + \frac{5}{12} = \pm \frac{\sqrt{97}}{\sqrt{12^{2}}}$

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

$m + \frac{5}{12} = \pm \frac{\sqrt{97}}{12}$

Subtract $\frac{5}{12}$ from both sides of the equation.

$m = \pm \frac{\sqrt{97}}{12} - \frac{5}{12}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$m = \pm \frac{\sqrt{97}}{12} - \frac{5}{12}$

Decimal Form:

$m = 0.40407148 \dots, - 1.23740481 \dots$

Solve by Completing the Square 6m^2+5m-3=0

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