Solve by Completing the Square 3x^2-24x+27=0

$3 x^{2} - 24 x + 27 = 0$

Step 1

Subtract $27$ from both sides of the equation.

$3 x^{2} - 24 x = - 27$

Step 2

Divide each term in $3 x^{2} - 24 x = - 27$ by $3$ and simplify.

Divide each term in $3 x^{2} - 24 x = - 27$ by $3$ .

$\frac{3 x^{2}}{3} + \frac{- 24 x}{3} = \frac{- 27}{3}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{- 24 x}{3} = \frac{- 27}{3}$

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

$x^{2} + \frac{- 24 x}{3} = \frac{- 27}{3}$

Cancel the common factor of $- 24$ and $3$ .

Factor $3$ out of $- 24 x$ .

$x^{2} + \frac{3 (- 8 x)}{3} = \frac{- 27}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$x^{2} + \frac{3 (- 8 x)}{3 (1)} = \frac{- 27}{3}$

Cancel the common factor.

$x^{2} + \frac{3 (- 8 x)}{3 \cdot 1} = \frac{- 27}{3}$

Rewrite the expression.

$x^{2} + \frac{- 8 x}{1} = \frac{- 27}{3}$

Divide $- 8 x$ by $1$ .

$x^{2} - 8 x = \frac{- 27}{3}$

Simplify the right side.

Divide $- 27$ by $3$ .

$x^{2} - 8 x = - 9$

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 $b$ .

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

Step 4

Add the term to each side of the equation.

$x^{2} - 8 x + {(- 4)}^{2} = - 9 + {(- 4)}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

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

$x^{2} - 8 x + 16 = - 9 + {(- 4)}^{2}$

Simplify the right side.

Simplify $- 9 + {(- 4)}^{2}$ .

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

$x^{2} - 8 x + 16 = - 9 + 16$

Add $- 9$ and $16$ .

$x^{2} - 8 x + 16 = 7$

Step 6

Factor the perfect trinomial square into ${(x - 4)}^{2}$ .

${(x - 4)}^{2} = 7$

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.

$x - 4 = \pm \sqrt{7}$

Add $4$ to both sides of the equation.

$x = \pm \sqrt{7} + 4$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \sqrt{7} + 4$

Decimal Form:

$x = 6.64575131 \dots, 1.35424868 \dots$

Solve by Completing the Square 3x^2-24x+27=0

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