Find All Complex Solutions (5y+6)^2=24

${(5 y + 6)}^{2} = 24$

Step 1

Move all terms to the left side of the equation and simplify.

Subtract $24$ from both sides of the equation.

${(5 y + 6)}^{2} - 24 = 0$

Simplify ${(5 y + 6)}^{2} - 24$ .

Simplify each term.

Rewrite ${(5 y + 6)}^{2}$ as $(5 y + 6) (5 y + 6)$ .

$(5 y + 6) (5 y + 6) - 24 = 0$

Expand $(5 y + 6) (5 y + 6)$ using the FOIL Method.

Apply the distributive property.

$5 y (5 y + 6) + 6 (5 y + 6) - 24 = 0$

Apply the distributive property.

$5 y (5 y) + 5 y \cdot 6 + 6 (5 y + 6) - 24 = 0$

Apply the distributive property.

$5 y (5 y) + 5 y \cdot 6 + 6 (5 y) + 6 \cdot 6 - 24 = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$5 \cdot 5 y \cdot y + 5 y \cdot 6 + 6 (5 y) + 6 \cdot 6 - 24 = 0$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$5 \cdot 5 (y \cdot y) + 5 y \cdot 6 + 6 (5 y) + 6 \cdot 6 - 24 = 0$

Multiply $y$ by $y$ .

$5 \cdot 5 y^{2} + 5 y \cdot 6 + 6 (5 y) + 6 \cdot 6 - 24 = 0$

Multiply $5$ by $5$ .

$25 y^{2} + 5 y \cdot 6 + 6 (5 y) + 6 \cdot 6 - 24 = 0$

Multiply $6$ by $5$ .

$25 y^{2} + 30 y + 6 (5 y) + 6 \cdot 6 - 24 = 0$

Multiply $5$ by $6$ .

$25 y^{2} + 30 y + 30 y + 6 \cdot 6 - 24 = 0$

Multiply $6$ by $6$ .

$25 y^{2} + 30 y + 30 y + 36 - 24 = 0$

Add $30 y$ and $30 y$ .

$25 y^{2} + 60 y + 36 - 24 = 0$

Subtract $24$ from $36$ .

$25 y^{2} + 60 y + 12 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 25$ , $b = 60$ , and $c = 12$ into the quadratic formula and solve for $y$ .

$\frac{- 60 \pm \sqrt{60^{2} - 4 \cdot (25 \cdot 12)}}{2 \cdot 25}$

Step 4

Simplify.

Simplify the numerator.

Raise $60$ to the power of $2$ .

$y = \frac{- 60 \pm \sqrt{3600 - 4 \cdot 25 \cdot 12}}{2 \cdot 25}$

Multiply $- 4 \cdot 25 \cdot 12$ .

Multiply $- 4$ by $25$ .

$y = \frac{- 60 \pm \sqrt{3600 - 100 \cdot 12}}{2 \cdot 25}$

Multiply $- 100$ by $12$ .

$y = \frac{- 60 \pm \sqrt{3600 - 1200}}{2 \cdot 25}$

Subtract $1200$ from $3600$ .

$y = \frac{- 60 \pm \sqrt{2400}}{2 \cdot 25}$

Rewrite $2400$ as $20^{2} \cdot 6$ .

Factor $400$ out of $2400$ .

$y = \frac{- 60 \pm \sqrt{400 (6)}}{2 \cdot 25}$

Rewrite $400$ as $20^{2}$ .

$y = \frac{- 60 \pm \sqrt{20^{2} \cdot 6}}{2 \cdot 25}$

Pull terms out from under the radical.

$y = \frac{- 60 \pm 20 \sqrt{6}}{2 \cdot 25}$

Multiply $2$ by $25$ .

$y = \frac{- 60 \pm 20 \sqrt{6}}{50}$

Simplify $\frac{- 60 \pm 20 \sqrt{6}}{50}$ .

$y = \frac{- 6 \pm 2 \sqrt{6}}{5}$

Step 5

The final answer is the combination of both solutions.

$y = - \frac{6 - 2 \sqrt{6}}{5}, - \frac{6 + 2 \sqrt{6}}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{6 - 2 \sqrt{6}}{5}, - \frac{6 + 2 \sqrt{6}}{5}$

Decimal Form:

$y = - 0.22020410 \dots, - 2.17979589 \dots$

Find All Complex Solutions (5y+6)^2=24

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