Find the Roots (Zeros) y=-6(x-5)^2+12

$y = - 6 {(x - 5)}^{2} + 12$

Step 1

Set $- 6 {(x - 5)}^{2} + 12$ equal to $0$ .

$- 6 {(x - 5)}^{2} + 12 = 0$

Step 2

Solve for $x$ .

Simplify $- 6 {(x - 5)}^{2} + 12$ .

Simplify each term.

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$- 6 ((x - 5) (x - 5)) + 12 = 0$

Expand $(x - 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$- 6 (x (x - 5) - 5 (x - 5)) + 12 = 0$

Apply the distributive property.

$- 6 (x \cdot x + x \cdot - 5 - 5 (x - 5)) + 12 = 0$

Apply the distributive property.

$- 6 (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) + 12 = 0$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$- 6 (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) + 12 = 0$

Move $- 5$ to the left of $x$ .

$- 6 (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) + 12 = 0$

Multiply $- 5$ by $- 5$ .

$- 6 (x^{2} - 5 x - 5 x + 25) + 12 = 0$

Subtract $5 x$ from $- 5 x$ .

$- 6 (x^{2} - 10 x + 25) + 12 = 0$

Apply the distributive property.

$- 6 x^{2} - 6 (- 10 x) - 6 \cdot 25 + 12 = 0$

Simplify.

Multiply $- 10$ by $- 6$ .

$- 6 x^{2} + 60 x - 6 \cdot 25 + 12 = 0$

Multiply $- 6$ by $25$ .

$- 6 x^{2} + 60 x - 150 + 12 = 0$

Add $- 150$ and $12$ .

$- 6 x^{2} + 60 x - 138 = 0$

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 5 - \sqrt{2}, 5 + \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 5 - \sqrt{2}, 5 + \sqrt{2}$

Decimal Form:

$x = 3.58578643 \dots, 6.41421356 \dots$

Step 4