Find the Roots (Zeros) 9x^2-1-(3x-2)^2=0

$9 x^{2} - 1 - {(3 x - 2)}^{2} = 0$

Step 1

Simplify $9 x^{2} - 1 - {(3 x - 2)}^{2}$ .

Simplify each term.

Rewrite ${(3 x - 2)}^{2}$ as $(3 x - 2) (3 x - 2)$ .

$9 x^{2} - 1 - ((3 x - 2) (3 x - 2)) = 0$

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

Apply the distributive property.

$9 x^{2} - 1 - (3 x (3 x - 2) - 2 (3 x - 2)) = 0$

Apply the distributive property.

$9 x^{2} - 1 - (3 x (3 x) + 3 x \cdot - 2 - 2 (3 x - 2)) = 0$

Apply the distributive property.

$9 x^{2} - 1 - (3 x (3 x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$9 x^{2} - 1 - (3 \cdot 3 x \cdot x + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

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

Move $x$ .

$9 x^{2} - 1 - (3 \cdot 3 (x \cdot x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

Multiply $x$ by $x$ .

$9 x^{2} - 1 - (3 \cdot 3 x^{2} + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

Multiply $3$ by $3$ .

$9 x^{2} - 1 - (9 x^{2} + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) = 0$

Multiply $- 2$ by $3$ .

$9 x^{2} - 1 - (9 x^{2} - 6 x - 2 (3 x) - 2 \cdot - 2) = 0$

Multiply $3$ by $- 2$ .

$9 x^{2} - 1 - (9 x^{2} - 6 x - 6 x - 2 \cdot - 2) = 0$

Multiply $- 2$ by $- 2$ .

$9 x^{2} - 1 - (9 x^{2} - 6 x - 6 x + 4) = 0$

Subtract $6 x$ from $- 6 x$ .

$9 x^{2} - 1 - (9 x^{2} - 12 x + 4) = 0$

Apply the distributive property.

$9 x^{2} - 1 - (9 x^{2}) - (- 12 x) - 1 \cdot 4 = 0$

Simplify.

Multiply $9$ by $- 1$ .

$9 x^{2} - 1 - 9 x^{2} - (- 12 x) - 1 \cdot 4 = 0$

Multiply $- 12$ by $- 1$ .

$9 x^{2} - 1 - 9 x^{2} + 12 x - 1 \cdot 4 = 0$

Multiply $- 1$ by $4$ .

$9 x^{2} - 1 - 9 x^{2} + 12 x - 4 = 0$

Simplify by adding terms.

Combine the opposite terms in $9 x^{2} - 1 - 9 x^{2} + 12 x - 4$ .

Subtract $9 x^{2}$ from $9 x^{2}$ .

$0 - 1 + 12 x - 4 = 0$

Subtract $1$ from $0$ .

$- 1 + 12 x - 4 = 0$

Subtract $4$ from $- 1$ .

$12 x - 5 = 0$

Step 2

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

$x \approx 0.41666666$

Step 3

Find the Roots (Zeros) 9x^2-1-(3x-2)^2=0

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