Solve by Completing the Square 3x^2+28x-3=0

$3 x^{2} + 28 x - 3 = 0$

Step 1

Add $3$ to both sides of the equation.

$3 x^{2} + 28 x = 3$

Step 2

Divide each term in $3 x^{2} + 28 x = 3$ by $3$ and simplify.

Divide each term in $3 x^{2} + 28 x = 3$ by $3$ .

$\frac{3 x^{2}}{3} + \frac{28 x}{3} = \frac{3}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{28 x}{3} = \frac{3}{3}$

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

$x^{2} + \frac{28 x}{3} = \frac{3}{3}$

Simplify the right side.

Divide $3$ by $3$ .

$x^{2} + \frac{28 x}{3} = 1$

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{14}{3})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{28 x}{3} + {(\frac{14}{3})}^{2} = 1 + {(\frac{14}{3})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{14}{3}$ .

$x^{2} + \frac{28 x}{3} + \frac{14^{2}}{3^{2}} = 1 + {(\frac{14}{3})}^{2}$

Raise $14$ to the power of $2$ .

$x^{2} + \frac{28 x}{3} + \frac{196}{3^{2}} = 1 + {(\frac{14}{3})}^{2}$

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

$x^{2} + \frac{28 x}{3} + \frac{196}{9} = 1 + {(\frac{14}{3})}^{2}$

Simplify the right side.

Simplify $1 + {(\frac{14}{3})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{14}{3}$ .

$x^{2} + \frac{28 x}{3} + \frac{196}{9} = 1 + \frac{14^{2}}{3^{2}}$

Raise $14$ to the power of $2$ .

$x^{2} + \frac{28 x}{3} + \frac{196}{9} = 1 + \frac{196}{3^{2}}$

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

$x^{2} + \frac{28 x}{3} + \frac{196}{9} = 1 + \frac{196}{9}$

Simplify the expression.

Write $1$ as a fraction with a common denominator.

$x^{2} + \frac{28 x}{3} + \frac{196}{9} = \frac{9}{9} + \frac{196}{9}$

Combine the numerators over the common denominator.

$x^{2} + \frac{28 x}{3} + \frac{196}{9} = \frac{9 + 196}{9}$

Add $9$ and $196$ .

$x^{2} + \frac{28 x}{3} + \frac{196}{9} = \frac{205}{9}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{14}{3})}^{2}$ .

${(x + \frac{14}{3})}^{2} = \frac{205}{9}$

Step 7

Solve the equation for $x$ .

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

$x + \frac{14}{3} = \pm \sqrt{\frac{205}{9}}$

Simplify $\pm \sqrt{\frac{205}{9}}$ .

Rewrite $\sqrt{\frac{205}{9}}$ as $\frac{\sqrt{205}}{\sqrt{9}}$ .

$x + \frac{14}{3} = \pm \frac{\sqrt{205}}{\sqrt{9}}$

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$x + \frac{14}{3} = \pm \frac{\sqrt{205}}{\sqrt{3^{2}}}$

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

$x + \frac{14}{3} = \pm \frac{\sqrt{205}}{3}$

Subtract $\frac{14}{3}$ from both sides of the equation.

$x = \pm \frac{\sqrt{205}}{3} - \frac{14}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{205}}{3} - \frac{14}{3}$

Decimal Form:

$x = 0.10594035 \dots, - 9.43927368 \dots$

Solve by Completing the Square 3x^2+28x-3=0

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