Solve Using the Quadratic Formula 2(s+1)^2-5(s+1)=3

$2 {(s + 1)}^{2} - 5 (s + 1) = 3$

Step 1

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

Simplify the left side.

Simplify each term.

Apply the distributive property.

$2 {(s + 1)}^{2} - 5 s - 5 \cdot 1 = 3$

Multiply $- 5$ by $1$ .

$2 {(s + 1)}^{2} - 5 s - 5 = 3$

Subtract $3$ from both sides of the equation.

$2 {(s + 1)}^{2} - 5 s - 5 - 3 = 0$

Simplify $2 {(s + 1)}^{2} - 5 s - 5 - 3$ .

Simplify each term.

Rewrite ${(s + 1)}^{2}$ as $(s + 1) (s + 1)$ .

$2 ((s + 1) (s + 1)) - 5 s - 5 - 3 = 0$

Expand $(s + 1) (s + 1)$ using the FOIL Method.

Apply the distributive property.

$2 (s (s + 1) + 1 (s + 1)) - 5 s - 5 - 3 = 0$

Apply the distributive property.

$2 (s \cdot s + s \cdot 1 + 1 (s + 1)) - 5 s - 5 - 3 = 0$

Apply the distributive property.

$2 (s \cdot s + s \cdot 1 + 1 s + 1 \cdot 1) - 5 s - 5 - 3 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $s$ by $s$ .

$2 (s^{2} + s \cdot 1 + 1 s + 1 \cdot 1) - 5 s - 5 - 3 = 0$

Multiply $s$ by $1$ .

$2 (s^{2} + s + 1 s + 1 \cdot 1) - 5 s - 5 - 3 = 0$

Multiply $s$ by $1$ .

$2 (s^{2} + s + s + 1 \cdot 1) - 5 s - 5 - 3 = 0$

Multiply $1$ by $1$ .

$2 (s^{2} + s + s + 1) - 5 s - 5 - 3 = 0$

Add $s$ and $s$ .

$2 (s^{2} + 2 s + 1) - 5 s - 5 - 3 = 0$

Apply the distributive property.

$2 s^{2} + 2 (2 s) + 2 \cdot 1 - 5 s - 5 - 3 = 0$

Simplify.

Multiply $2$ by $2$ .

$2 s^{2} + 4 s + 2 \cdot 1 - 5 s - 5 - 3 = 0$

Multiply $2$ by $1$ .

$2 s^{2} + 4 s + 2 - 5 s - 5 - 3 = 0$

Subtract $5 s$ from $4 s$ .

$2 s^{2} - s + 2 - 5 - 3 = 0$

Subtract $5$ from $2$ .

$2 s^{2} - s - 3 - 3 = 0$

Subtract $3$ from $- 3$ .

$2 s^{2} - s - 6 = 0$

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 $a = 2$ , $b = - 1$ , and $c = - 6$ into the quadratic formula and solve for $s$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (2 \cdot - 6)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 1$ to the power of $2$ .

$s = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 6}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 6$ .

Multiply $- 4$ by $2$ .

$s = \frac{1 \pm \sqrt{1 - 8 \cdot - 6}}{2 \cdot 2}$

Multiply $- 8$ by $- 6$ .

$s = \frac{1 \pm \sqrt{1 + 48}}{2 \cdot 2}$

Add $1$ and $48$ .

$s = \frac{1 \pm \sqrt{49}}{2 \cdot 2}$

Rewrite $49$ as $7^{2}$ .

$s = \frac{1 \pm \sqrt{7^{2}}}{2 \cdot 2}$

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

$s = \frac{1 \pm 7}{2 \cdot 2}$

Multiply $2$ by $2$ .

$s = \frac{1 \pm 7}{4}$

Step 5

The final answer is the combination of both solutions.

$s = 2, - \frac{3}{2}$

Solve Using the Quadratic Formula 2(s+1)^2-5(s+1)=3

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