Solve by Completing the Square 3x^2+15x-3=0

$3 x^{2} + 15 x - 3 = 0$

Step 1

Add $3$ to both sides of the equation.

$3 x^{2} + 15 x = 3$

Step 2

Divide each term in $3 x^{2} + 15 x = 3$ by $3$ and simplify.

Divide each term in $3 x^{2} + 15 x = 3$ by $3$ .

$\frac{3 x^{2}}{3} + \frac{15 x}{3} = \frac{3}{3}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{15 x}{3} = \frac{3}{3}$

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

$x^{2} + \frac{15 x}{3} = \frac{3}{3}$

Cancel the common factor of $15$ and $3$ .

Factor $3$ out of $15 x$ .

$x^{2} + \frac{3 (5 x)}{3} = \frac{3}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

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

Cancel the common factor.

$x^{2} + \frac{3 (5 x)}{3 \cdot 1} = \frac{3}{3}$

Rewrite the expression.

$x^{2} + \frac{5 x}{1} = \frac{3}{3}$

Divide $5 x$ by $1$ .

$x^{2} + 5 x = \frac{3}{3}$

Simplify the right side.

Divide $3$ by $3$ .

$x^{2} + 5 x = 1$

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 $b$ .

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

Step 4

Add the term to each side of the equation.

$x^{2} + 5 x + {(\frac{5}{2})}^{2} = 1 + {(\frac{5}{2})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{5}{2}$ .

$x^{2} + 5 x + \frac{5^{2}}{2^{2}} = 1 + {(\frac{5}{2})}^{2}$

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

$x^{2} + 5 x + \frac{25}{2^{2}} = 1 + {(\frac{5}{2})}^{2}$

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

$x^{2} + 5 x + \frac{25}{4} = 1 + {(\frac{5}{2})}^{2}$

Simplify the right side.

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

Simplify each term.

Apply the product rule to $\frac{5}{2}$ .

$x^{2} + 5 x + \frac{25}{4} = 1 + \frac{5^{2}}{2^{2}}$

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

$x^{2} + 5 x + \frac{25}{4} = 1 + \frac{25}{2^{2}}$

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

$x^{2} + 5 x + \frac{25}{4} = 1 + \frac{25}{4}$

Simplify the expression.

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

$x^{2} + 5 x + \frac{25}{4} = \frac{4}{4} + \frac{25}{4}$

Combine the numerators over the common denominator.

$x^{2} + 5 x + \frac{25}{4} = \frac{4 + 25}{4}$

Add $4$ and $25$ .

$x^{2} + 5 x + \frac{25}{4} = \frac{29}{4}$

Step 6

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

${(x + \frac{5}{2})}^{2} = \frac{29}{4}$

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.

$x + \frac{5}{2} = \pm \sqrt{\frac{29}{4}}$

Simplify $\pm \sqrt{\frac{29}{4}}$ .

Rewrite $\sqrt{\frac{29}{4}}$ as $\frac{\sqrt{29}}{\sqrt{4}}$ .

$x + \frac{5}{2} = \pm \frac{\sqrt{29}}{\sqrt{4}}$

Simplify the denominator.

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

$x + \frac{5}{2} = \pm \frac{\sqrt{29}}{\sqrt{2^{2}}}$

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

$x + \frac{5}{2} = \pm \frac{\sqrt{29}}{2}$

Subtract $\frac{5}{2}$ from both sides of the equation.

$x = \pm \frac{\sqrt{29}}{2} - \frac{5}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{29}}{2} - \frac{5}{2}$

Decimal Form:

$x = 0.19258240 \dots, - 5.19258240 \dots$

Solve by Completing the Square 3x^2+15x-3=0

Related Questions

Support LCMGCF.com by disabling your adblocker.