Solve by Completing the Square 4x^2+24x=-35

$4 x^{2} + 24 x = - 35$

Step 1

Divide each term in $4 x^{2} + 24 x = - 35$ by $4$ and simplify.

Divide each term in $4 x^{2} + 24 x = - 35$ by $4$ .

$\frac{4 x^{2}}{4} + \frac{24 x}{4} = \frac{- 35}{4}$

Simplify the left side.

Simplify each term.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} + \frac{24 x}{4} = \frac{- 35}{4}$

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

$x^{2} + \frac{24 x}{4} = \frac{- 35}{4}$

Cancel the common factor of $24$ and $4$ .

Factor $4$ out of $24 x$ .

$x^{2} + \frac{4 (6 x)}{4} = \frac{- 35}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$x^{2} + \frac{4 (6 x)}{4 (1)} = \frac{- 35}{4}$

Cancel the common factor.

$x^{2} + \frac{4 (6 x)}{4 \cdot 1} = \frac{- 35}{4}$

Rewrite the expression.

$x^{2} + \frac{6 x}{1} = \frac{- 35}{4}$

Divide $6 x$ by $1$ .

$x^{2} + 6 x = \frac{- 35}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} + 6 x = - \frac{35}{4}$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

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

Step 3

Add the term to each side of the equation.

$x^{2} + 6 x + {(3)}^{2} = - \frac{35}{4} + {(3)}^{2}$

Step 4

Simplify the equation.

Simplify the left side.

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

$x^{2} + 6 x + 9 = - \frac{35}{4} + {(3)}^{2}$

Simplify the right side.

Simplify $- \frac{35}{4} + {(3)}^{2}$ .

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

$x^{2} + 6 x + 9 = - \frac{35}{4} + 9$

To write $9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} + 6 x + 9 = - \frac{35}{4} + 9 \cdot \frac{4}{4}$

Combine $9$ and $\frac{4}{4}$ .

$x^{2} + 6 x + 9 = - \frac{35}{4} + \frac{9 \cdot 4}{4}$

Combine the numerators over the common denominator.

$x^{2} + 6 x + 9 = \frac{- 35 + 9 \cdot 4}{4}$

Simplify the numerator.

Multiply $9$ by $4$ .

$x^{2} + 6 x + 9 = \frac{- 35 + 36}{4}$

Add $- 35$ and $36$ .

$x^{2} + 6 x + 9 = \frac{1}{4}$

Step 5

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

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

Step 6

Solve the equation for $x$ .

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

$x + 3 = \pm \sqrt{\frac{1}{4}}$

Simplify $\pm \sqrt{\frac{1}{4}}$ .

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x + 3 = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Any root of $1$ is $1$ .

$x + 3 = \pm \frac{1}{\sqrt{4}}$

Simplify the denominator.

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

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

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

$x + 3 = \pm \frac{1}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

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

Move all terms not containing $x$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$x = \frac{1}{2} - 3$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = \frac{1}{2} - 3 \cdot \frac{2}{2}$

Combine $- 3$ and $\frac{2}{2}$ .

$x = \frac{1}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$x = \frac{1 - 3 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$x = \frac{1 - 6}{2}$

Subtract $6$ from $1$ .

$x = \frac{- 5}{2}$

Move the negative in front of the fraction.

$x = - \frac{5}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

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

Move all terms not containing $x$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$x = - \frac{1}{2} - 3$

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = - \frac{1}{2} - 3 \cdot \frac{2}{2}$

Combine $- 3$ and $\frac{2}{2}$ .

$x = - \frac{1}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$x = \frac{- 1 - 3 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 3$ by $2$ .

$x = \frac{- 1 - 6}{2}$

Subtract $6$ from $- 1$ .

$x = \frac{- 7}{2}$

Move the negative in front of the fraction.

$x = - \frac{7}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = - \frac{5}{2}, - \frac{7}{2}$

Solve by Completing the Square 4x^2+24x=-35

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