Solve Using the Quadratic Formula 2x=2x-x(6-x)+9

$2 x = 2 x - x (6 - x) + 9$

Step 1

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

Simplify the right side.

Simplify $2 x - x (6 - x) + 9$ .

Simplify each term.

Apply the distributive property.

$2 x = 2 x - x \cdot 6 - x (- x) + 9$

Multiply $6$ by $- 1$ .

$2 x = 2 x - 6 x - x (- x) + 9$

Rewrite using the commutative property of multiplication.

$2 x = 2 x - 6 x - 1 \cdot - 1 x \cdot x + 9$

Simplify each term.

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

Move $x$ .

$2 x = 2 x - 6 x - 1 \cdot - 1 (x \cdot x) + 9$

Multiply $x$ by $x$ .

$2 x = 2 x - 6 x - 1 \cdot - 1 x^{2} + 9$

Multiply $- 1$ by $- 1$ .

$2 x = 2 x - 6 x + 1 x^{2} + 9$

Multiply $x^{2}$ by $1$ .

$2 x = 2 x - 6 x + x^{2} + 9$

Subtract $6 x$ from $2 x$ .

$2 x = - 4 x + x^{2} + 9$

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

Add $4 x$ to both sides of the equation.

$2 x + 4 x = x^{2} + 9$

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

$2 x + 4 x - x^{2} = 9$

Subtract $9$ from both sides of the equation.

$2 x + 4 x - x^{2} - 9 = 0$

Add $2 x$ and $4 x$ .

$6 x - x^{2} - 9 = 0$

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 $a = - 1$ , $b = 6$ , and $c = - 9$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (- 1 \cdot - 9)}}{2 \cdot - 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot - 1 \cdot - 9}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot - 9$ .

Multiply $- 4$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 4 \cdot - 9}}{2 \cdot - 1}$

Multiply $4$ by $- 9$ .

$x = \frac{- 6 \pm \sqrt{36 - 36}}{2 \cdot - 1}$

Subtract $36$ from $36$ .

$x = \frac{- 6 \pm \sqrt{0}}{2 \cdot - 1}$

Rewrite $0$ as $0^{2}$ .

$x = \frac{- 6 \pm \sqrt{0^{2}}}{2 \cdot - 1}$

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

$x = \frac{- 6 \pm 0}{2 \cdot - 1}$

$- 6$ plus or minus $0$ is $- 6$ .

$x = \frac{- 6}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 6}{- 2}$

Divide $- 6$ by $- 2$ .

$x = 3$

Step 5

The final answer is the combination of both solutions.

$x = 3$ Double roots

Solve Using the Quadratic Formula 2x=2x-x(6-x)+9

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