Solve for x (2x-7)(4x+3)=112

$(2 x - 7) (4 x + 3) = 112$

Simplify $(2 x - 7) (4 x + 3)$ .

Expand $(2 x - 7) (4 x + 3)$ using the FOIL Method.

Apply the distributive property.

$2 x (4 x + 3) - 7 (4 x + 3) = 112$

Apply the distributive property.

$2 x (4 x) + 2 x \cdot 3 - 7 (4 x + 3) = 112$

Apply the distributive property.

$2 x (4 x) + 2 x \cdot 3 - 7 (4 x) - 7 \cdot 3 = 112$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 4 x \cdot x + 2 x \cdot 3 - 7 (4 x) - 7 \cdot 3 = 112$

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

Move $x$ .

$2 \cdot 4 (x \cdot x) + 2 x \cdot 3 - 7 (4 x) - 7 \cdot 3 = 112$

Multiply $x$ by $x$ .

$2 \cdot 4 x^{2} + 2 x \cdot 3 - 7 (4 x) - 7 \cdot 3 = 112$

Multiply $2$ by $4$ .

$8 x^{2} + 2 x \cdot 3 - 7 (4 x) - 7 \cdot 3 = 112$

Multiply $3$ by $2$ .

$8 x^{2} + 6 x - 7 (4 x) - 7 \cdot 3 = 112$

Multiply $4$ by $- 7$ .

$8 x^{2} + 6 x - 28 x - 7 \cdot 3 = 112$

Multiply $- 7$ by $3$ .

$8 x^{2} + 6 x - 28 x - 21 = 112$

Subtract $28 x$ from $6 x$ .

$8 x^{2} - 22 x - 21 = 112$

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

Subtract $112$ from both sides of the equation.

$8 x^{2} - 22 x - 21 - 112 = 0$

Subtract $112$ from $- 21$ .

$8 x^{2} - 22 x - 133 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 8$ , $b = - 22$ , and $c = - 133$ into the quadratic formula and solve for $x$ .

$\frac{22 \pm \sqrt{{(- 22)}^{2} - 4 \cdot (8 \cdot - 133)}}{2 \cdot 8}$

Simplify.

Simplify the numerator.

Raise $- 22$ to the power of $2$ .

$x = \frac{22 \pm \sqrt{484 - 4 \cdot 8 \cdot - 133}}{2 \cdot 8}$

Multiply $- 4$ by $8$ .

$x = \frac{22 \pm \sqrt{484 - 32 \cdot - 133}}{2 \cdot 8}$

Multiply $- 32$ by $- 133$ .

$x = \frac{22 \pm \sqrt{484 + 4256}}{2 \cdot 8}$

Add $484$ and $4256$ .

$x = \frac{22 \pm \sqrt{4740}}{2 \cdot 8}$

Rewrite $4740$ as $2^{2} \cdot 1185$ .

Factor $4$ out of $4740$ .

$x = \frac{22 \pm \sqrt{4 (1185)}}{2 \cdot 8}$

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

$x = \frac{22 \pm \sqrt{2^{2} \cdot 1185}}{2 \cdot 8}$

Pull terms out from under the radical.

$x = \frac{22 \pm 2 \sqrt{1185}}{2 \cdot 8}$

Multiply $2$ by $8$ .

$x = \frac{22 \pm 2 \sqrt{1185}}{16}$

Simplify $\frac{22 \pm 2 \sqrt{1185}}{16}$ .

$x = \frac{11 \pm \sqrt{1185}}{8}$

The final answer is the combination of both solutions.

$x = \frac{11 + \sqrt{1185}}{8}, \frac{11 - \sqrt{1185}}{8}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11 + \sqrt{1185}}{8}, \frac{11 - \sqrt{1185}}{8}$

Decimal Form:

$x = 5.67797861 \dots, - 2.92797861 \dots$

Solve for x (2x-7)(4x+3)=112

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