Solve Using the Quadratic Formula 4/9x^2-4/3x=-1

$\frac{4}{9} x^{2} - \frac{4}{3} x = - 1$

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

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Simplify each term.

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Combine $\frac{4}{9}$ and $x^{2}$ .

$\frac{4 x^{2}}{9} - \frac{4}{3} x = - 1$

Combine $x$ and $\frac{4}{3}$ .

$\frac{4 x^{2}}{9} - \frac{x \cdot 4}{3} = - 1$

Move $4$ to the left of $x$ .

$\frac{4 x^{2}}{9} - \frac{4 x}{3} = - 1$

Add $1$ to both sides of the equation.

$\frac{4 x^{2}}{9} - \frac{4 x}{3} + 1 = 0$

Multiply through by the least common denominator $9$ , then simplify.

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Apply the distributive property.

$9 (\frac{4 x^{2}}{9}) + 9 (- \frac{4 x}{3}) + 9 \cdot 1 = 0$

Simplify.

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Cancel the common factor of $9$ .

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Cancel the common factor.

$9 (\frac{4 x^{2}}{9}) + 9 (- \frac{4 x}{3}) + 9 \cdot 1 = 0$

Rewrite the expression.

$4 x^{2} + 9 (- \frac{4 x}{3}) + 9 \cdot 1 = 0$

Cancel the common factor of $3$ .

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Move the leading negative in $- \frac{4 x}{3}$ into the numerator.

$4 x^{2} + 9 (\frac{- 4 x}{3}) + 9 \cdot 1 = 0$

Factor $3$ out of $9$ .

$4 x^{2} + 3 (3) (\frac{- 4 x}{3}) + 9 \cdot 1 = 0$

Cancel the common factor.

$4 x^{2} + 3 \cdot (3 (\frac{- 4 x}{3})) + 9 \cdot 1 = 0$

Rewrite the expression.

$4 x^{2} + 3 (- 4 x) + 9 \cdot 1 = 0$

Multiply $- 4$ by $3$ .

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

Multiply $9$ by $1$ .

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 4$ , $b = - 12$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{12 \pm \sqrt{{(- 12)}^{2} - 4 \cdot (4 \cdot 9)}}{2 \cdot 4}$

Simplify.

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Simplify the numerator.

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

$x = \frac{12 \pm \sqrt{144 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Multiply $- 4$ by $4$ .

$x = \frac{12 \pm \sqrt{144 - 16 \cdot 9}}{2 \cdot 4}$

Multiply $- 16$ by $9$ .

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

Subtract $144$ from $144$ .

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

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

$x = \frac{12 \pm \sqrt{0^{2}}}{2 \cdot 4}$

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

$x = \frac{12 \pm 0}{2 \cdot 4}$

Multiply $2$ by $4$ .

$x = \frac{12 \pm 0}{8}$

Simplify $\frac{12 \pm 0}{8}$ .

$x = \frac{12}{8}$

Cancel the common factor of $12$ and $8$ .

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Factor $4$ out of $12$ .

$x = \frac{4 (3)}{8}$

Cancel the common factors.

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Factor $4$ out of $8$ .

$x = \frac{4 \cdot 3}{4 \cdot 2}$

Cancel the common factor.

$x = \frac{4 \cdot 3}{4 \cdot 2}$

Rewrite the expression.

$x = \frac{3}{2}$

The final answer is the combination of both solutions.

$x = \frac{3}{2}$ Double roots

Solve Using the Quadratic Formula 4/9x^2-4/3x=-1

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