Solve Using the Quadratic Formula (2x-1)(3x+2)=x+292

$(2 x - 1) (3 x + 2) = x + 292$

Step 1

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

Simplify the left side.

Simplify $(2 x - 1) (3 x + 2)$ .

Expand $(2 x - 1) (3 x + 2)$ using the FOIL Method.

Apply the distributive property.

$2 x (3 x + 2) - 1 (3 x + 2) = x + 292$

Apply the distributive property.

$2 x (3 x) + 2 x \cdot 2 - 1 (3 x + 2) = x + 292$

Apply the distributive property.

$2 x (3 x) + 2 x \cdot 2 - 1 (3 x) - 1 \cdot 2 = x + 292$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 3 x \cdot x + 2 x \cdot 2 - 1 (3 x) - 1 \cdot 2 = x + 292$

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

Move $x$ .

$2 \cdot 3 (x \cdot x) + 2 x \cdot 2 - 1 (3 x) - 1 \cdot 2 = x + 292$

Multiply $x$ by $x$ .

$2 \cdot 3 x^{2} + 2 x \cdot 2 - 1 (3 x) - 1 \cdot 2 = x + 292$

Multiply $2$ by $3$ .

$6 x^{2} + 2 x \cdot 2 - 1 (3 x) - 1 \cdot 2 = x + 292$

Multiply $2$ by $2$ .

$6 x^{2} + 4 x - 1 (3 x) - 1 \cdot 2 = x + 292$

Multiply $3$ by $- 1$ .

$6 x^{2} + 4 x - 3 x - 1 \cdot 2 = x + 292$

Multiply $- 1$ by $2$ .

$6 x^{2} + 4 x - 3 x - 2 = x + 292$

Subtract $3 x$ from $4 x$ .

$6 x^{2} + x - 2 = x + 292$

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

Subtract $x$ from both sides of the equation.

$6 x^{2} + x - 2 - x = 292$

Subtract $292$ from both sides of the equation.

$6 x^{2} + x - 2 - x - 292 = 0$

Simplify $6 x^{2} + x - 2 - x - 292$ .

Combine the opposite terms in $6 x^{2} + x - 2 - x - 292$ .

Subtract $x$ from $x$ .

$6 x^{2} + 0 - 2 - 292 = 0$

Add $6 x^{2}$ and $0$ .

$6 x^{2} - 2 - 292 = 0$

Subtract $292$ from $- 2$ .

$6 x^{2} - 294 = 0$

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 $a = 6$ , $b = 0$ , and $c = - 294$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (6 \cdot - 294)}}{2 \cdot 6}$

Step 4

Simplify.

Simplify the numerator.

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 6 \cdot - 294}}{2 \cdot 6}$

Multiply $- 4 \cdot 6 \cdot - 294$ .

Multiply $- 4$ by $6$ .

$x = \frac{0 \pm \sqrt{0 - 24 \cdot - 294}}{2 \cdot 6}$

Multiply $- 24$ by $- 294$ .

$x = \frac{0 \pm \sqrt{0 + 7056}}{2 \cdot 6}$

Add $0$ and $7056$ .

$x = \frac{0 \pm \sqrt{7056}}{2 \cdot 6}$

Rewrite $7056$ as $84^{2}$ .

$x = \frac{0 \pm \sqrt{84^{2}}}{2 \cdot 6}$

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

$x = \frac{0 \pm 84}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{0 \pm 84}{12}$

Simplify $\frac{0 \pm 84}{12}$ .

$x = \pm 7$

Step 5

The final answer is the combination of both solutions.

$x = 7, - 7$

Solve Using the Quadratic Formula (2x-1)(3x+2)=x+292

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