Solve for x (2x+3)^2+8(2x+3)+11=0

${(2 x + 3)}^{2} + 8 (2 x + 3) + 11 = 0$

Simplify ${(2 x + 3)}^{2} + 8 (2 x + 3) + 11$ .

Simplify each term.

Apply the distributive property.

${(2 x + 3)}^{2} + 8 (2 x) + 8 \cdot 3 + 11 = 0$

Multiply $2$ by $8$ .

${(2 x + 3)}^{2} + 16 x + 8 \cdot 3 + 11 = 0$

Multiply $8$ by $3$ .

${(2 x + 3)}^{2} + 16 x + 24 + 11 = 0$

Add $24$ and $11$ .

${(2 x + 3)}^{2} + 16 x + 35 = 0$

Factor the left side of the equation.

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$4 u^{2} + 28 u + 44 = 0$

Factor $4$ out of $4 u^{2} + 28 u + 44$ .

Factor $4$ out of $4 u^{2}$ .

$4 (u^{2}) + 28 u + 44 = 0$

Factor $4$ out of $28 u$ .

$4 (u^{2}) + 4 (7 u) + 44 = 0$

Factor $4$ out of $44$ .

$4 u^{2} + 4 (7 u) + 4 \cdot 11 = 0$

Factor $4$ out of $4 u^{2} + 4 (7 u)$ .

$4 (u^{2} + 7 u) + 4 \cdot 11 = 0$

Factor $4$ out of $4 (u^{2} + 7 u) + 4 \cdot 11$ .

$4 (u^{2} + 7 u + 11) = 0$

Replace all occurrences of $u$ with $x$ .

$4 (x^{2} + 7 x + 11) = 0$

Divide each term in $4 (x^{2} + 7 x + 11) = 0$ by $4$ and simplify.

Divide each term in $4 (x^{2} + 7 x + 11) = 0$ by $4$ .

$\frac{4 (x^{2} + 7 x + 11)}{4} = \frac{0}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 (x^{2} + 7 x + 11)}{4} = \frac{0}{4}$

Divide $x^{2} + 7 x + 11$ by $1$ .

$x^{2} + 7 x + 11 = \frac{0}{4}$

Simplify the right side.

Divide $0$ by $4$ .

$x^{2} + 7 x + 11 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 7$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (1 \cdot 11)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 1 \cdot 11}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot 11}}{2 \cdot 1}$

Multiply $- 4$ by $11$ .

$x = \frac{- 7 \pm \sqrt{49 - 44}}{2 \cdot 1}$

Subtract $44$ from $49$ .

$x = \frac{- 7 \pm \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 7 \pm \sqrt{5}}{2}$

The final answer is the combination of both solutions.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 2.38196601 \dots, - 4.61803398 \dots$

Solve for x (2x+3)^2+8(2x+3)+11=0

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