Solve Using the Quadratic Formula 4x^2-x+16+3x=7x-12-5x

$4 x^{2} - x + 16 + 3 x = 7 x - 12 - 5 x$

Step 1

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

Simplify the left side.

Add $- x$ and $3 x$ .

$4 x^{2} + 2 x + 16 = 7 x - 12 - 5 x$

Simplify the right side.

Subtract $5 x$ from $7 x$ .

$4 x^{2} + 2 x + 16 = 2 x - 12$

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

Subtract $2 x$ from both sides of the equation.

$4 x^{2} + 2 x + 16 - 2 x = - 12$

Add $12$ to both sides of the equation.

$4 x^{2} + 2 x + 16 - 2 x + 12 = 0$

Simplify $4 x^{2} + 2 x + 16 - 2 x + 12$ .

Combine the opposite terms in $4 x^{2} + 2 x + 16 - 2 x + 12$ .

Subtract $2 x$ from $2 x$ .

$4 x^{2} + 0 + 16 + 12 = 0$

Add $4 x^{2}$ and $0$ .

$4 x^{2} + 16 + 12 = 0$

Add $16$ and $12$ .

$4 x^{2} + 28 = 0$

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 $a = 4$ , $b = 0$ , and $c = 28$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (4 \cdot 28)}}{2 \cdot 4}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 4 \cdot 28}}{2 \cdot 4}$

Multiply $- 4 \cdot 4 \cdot 28$ .

Multiply $- 4$ by $4$ .

$x = \frac{0 \pm \sqrt{0 - 16 \cdot 28}}{2 \cdot 4}$

Multiply $- 16$ by $28$ .

$x = \frac{0 \pm \sqrt{0 - 448}}{2 \cdot 4}$

Subtract $448$ from $0$ .

$x = \frac{0 \pm \sqrt{- 448}}{2 \cdot 4}$

Rewrite $- 448$ as $- 1 (448)$ .

$x = \frac{0 \pm \sqrt{- 1 \cdot 448}}{2 \cdot 4}$

Rewrite $\sqrt{- 1 (448)}$ as $\sqrt{- 1} \cdot \sqrt{448}$ .

$x = \frac{0 \pm \sqrt{- 1} \cdot \sqrt{448}}{2 \cdot 4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{0 \pm i \cdot \sqrt{448}}{2 \cdot 4}$

Rewrite $448$ as $8^{2} \cdot 7$ .

Factor $64$ out of $448$ .

$x = \frac{0 \pm i \cdot \sqrt{64 (7)}}{2 \cdot 4}$

Rewrite $64$ as $8^{2}$ .

$x = \frac{0 \pm i \cdot \sqrt{8^{2} \cdot 7}}{2 \cdot 4}$

Pull terms out from under the radical.

$x = \frac{0 \pm i \cdot (8 \sqrt{7})}{2 \cdot 4}$

Move $8$ to the left of $i$ .

$x = \frac{0 \pm 8 i \sqrt{7}}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{0 \pm 8 i \sqrt{7}}{8}$

Simplify $\frac{0 \pm 8 i \sqrt{7}}{8}$ .

$x = \pm i \sqrt{7}$

Solve Using the Quadratic Formula 4x^2-x+16+3x=7x-12-5x

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