Solve Using the Quadratic Formula 5(x+3)-5(x^2-1)=x^2+7(3-x)-1

$5 (x + 3) - 5 (x^{2} - 1) = x^{2} + 7 (3 - x) - 1$

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

Simplify the left side.

Simplify $5 (x + 3) - 5 (x^{2} - 1)$ .

Simplify each term.

Apply the distributive property.

$5 x + 5 \cdot 3 - 5 (x^{2} - 1) = x^{2} + 7 (3 - x) - 1$

Multiply $5$ by $3$ .

$5 x + 15 - 5 (x^{2} - 1) = x^{2} + 7 (3 - x) - 1$

Apply the distributive property.

$5 x + 15 - 5 x^{2} - 5 \cdot - 1 = x^{2} + 7 (3 - x) - 1$

Multiply $- 5$ by $- 1$ .

$5 x + 15 - 5 x^{2} + 5 = x^{2} + 7 (3 - x) - 1$

Add $15$ and $5$ .

$5 x - 5 x^{2} + 20 = x^{2} + 7 (3 - x) - 1$

Simplify the right side.

Simplify $x^{2} + 7 (3 - x) - 1$ .

Simplify each term.

Apply the distributive property.

$5 x - 5 x^{2} + 20 = x^{2} + 7 \cdot 3 + 7 (- x) - 1$

Multiply $7$ by $3$ .

$5 x - 5 x^{2} + 20 = x^{2} + 21 + 7 (- x) - 1$

Multiply $- 1$ by $7$ .

$5 x - 5 x^{2} + 20 = x^{2} + 21 - 7 x - 1$

Subtract $1$ from $21$ .

$5 x - 5 x^{2} + 20 = x^{2} - 7 x + 20$

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

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

$5 x - 5 x^{2} + 20 - x^{2} = - 7 x + 20$

Add $7 x$ to both sides of the equation.

$5 x - 5 x^{2} + 20 - x^{2} + 7 x = 20$

Subtract $20$ from both sides of the equation.

$5 x - 5 x^{2} + 20 - x^{2} + 7 x - 20 = 0$

Simplify $5 x - 5 x^{2} + 20 - x^{2} + 7 x - 20$ .

Combine the opposite terms in $5 x - 5 x^{2} + 20 - x^{2} + 7 x - 20$ .

Subtract $20$ from $20$ .

$5 x - 5 x^{2} - x^{2} + 7 x + 0 = 0$

Add $5 x - 5 x^{2} - x^{2} + 7 x$ and $0$ .

$5 x - 5 x^{2} - x^{2} + 7 x = 0$

Add $5 x$ and $7 x$ .

$- 5 x^{2} - x^{2} + 12 x = 0$

Subtract $x^{2}$ from $- 5 x^{2}$ .

$- 6 x^{2} + 12 x = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 6$ , $b = 12$ , and $c = 0$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (- 6 \cdot 0)}}{2 \cdot - 6}$

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 6 \cdot 0}}{2 \cdot - 6}$

Multiply $- 4$ by $- 6$ .

$x = \frac{- 12 \pm \sqrt{144 + 24 \cdot 0}}{2 \cdot - 6}$

Multiply $24$ by $0$ .

$x = \frac{- 12 \pm \sqrt{144 + 0}}{2 \cdot - 6}$

Add $144$ and $0$ .

$x = \frac{- 12 \pm \sqrt{144}}{2 \cdot - 6}$

Rewrite $144$ as $12^{2}$ .

$x = \frac{- 12 \pm \sqrt{12^{2}}}{2 \cdot - 6}$

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

$x = \frac{- 12 \pm 12}{2 \cdot - 6}$

Multiply $2$ by $- 6$ .

$x = \frac{- 12 \pm 12}{- 12}$

Simplify $\frac{- 12 \pm 12}{- 12}$ .

$x = 1 \pm 1$

The final answer is the combination of both solutions.

$x = 2, 0$

Solve Using the Quadratic Formula 5(x+3)-5(x^2-1)=x^2+7(3-x)-1

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