Solve Using the Quadratic Formula 45=(2x+1)(x+1)

$45 = (2 x + 1) (x + 1)$

Step 1

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

Simplify the right side.

Simplify $(2 x + 1) (x + 1)$ .

Expand $(2 x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$45 = 2 x (x + 1) + 1 (x + 1)$

Apply the distributive property.

$45 = 2 x \cdot x + 2 x \cdot 1 + 1 (x + 1)$

Apply the distributive property.

$45 = 2 x \cdot x + 2 x \cdot 1 + 1 x + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$45 = 2 (x \cdot x) + 2 x \cdot 1 + 1 x + 1 \cdot 1$

Multiply $x$ by $x$ .

$45 = 2 x^{2} + 2 x \cdot 1 + 1 x + 1 \cdot 1$

Multiply $2$ by $1$ .

$45 = 2 x^{2} + 2 x + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$45 = 2 x^{2} + 2 x + x + 1 \cdot 1$

Multiply $1$ by $1$ .

$45 = 2 x^{2} + 2 x + x + 1$

Add $2 x$ and $x$ .

$45 = 2 x^{2} + 3 x + 1$

Move all the expressions to the left side of the equation.

Subtract $2 x^{2}$ from both sides of the equation.

$45 - 2 x^{2} = 3 x + 1$

Subtract $3 x$ from both sides of the equation.

$45 - 2 x^{2} - 3 x = 1$

Subtract $1$ from both sides of the equation.

$45 - 2 x^{2} - 3 x - 1 = 0$

Subtract $1$ from $45$ .

$- 2 x^{2} - 3 x + 44 = 0$

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 $a = - 2$ , $b = - 3$ , and $c = 44$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (- 2 \cdot 44)}}{2 \cdot - 2}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 2 \cdot 44}}{2 \cdot - 2}$

Multiply $- 4 \cdot - 2 \cdot 44$ .

Multiply $- 4$ by $- 2$ .

$x = \frac{3 \pm \sqrt{9 + 8 \cdot 44}}{2 \cdot - 2}$

Multiply $8$ by $44$ .

$x = \frac{3 \pm \sqrt{9 + 352}}{2 \cdot - 2}$

Add $9$ and $352$ .

$x = \frac{3 \pm \sqrt{361}}{2 \cdot - 2}$

Rewrite $361$ as $19^{2}$ .

$x = \frac{3 \pm \sqrt{19^{2}}}{2 \cdot - 2}$

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

$x = \frac{3 \pm 19}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{3 \pm 19}{- 4}$

Move the negative in front of the fraction.

$x = - \frac{3 \pm 19}{4}$

Step 5

The final answer is the combination of both solutions.

$x = - \frac{11}{2}, 4$

Solve Using the Quadratic Formula 45=(2x+1)(x+1)

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