Questions: Use the zero-factor property to solve the equation. 9 x^2 = 22 x - 8 The solutions set is . (Use a comma to separate answers. Type repeated roots only once.)

Use the zero-factor property to solve the equation.
9 x^2 = 22 x - 8

The solutions set is . (Use a comma to separate answers. Type repeated roots only once.)

Transcript text: Use the zero-factor property to solve the equation. \[ 9 x^{2}=22 x-8 \] The solutions set is $\square$ \}. (Use a comma to separate answers. Type repeated roots only once.)

Solution

Solution Steps

To solve the quadratic equation using the zero-factor property, we need to:

Move all terms to one side of the equation to set it to zero.
Factor the quadratic expression.
Set each factor equal to zero and solve for $ x $.

Solution Approach

Rewrite the equation in standard form: $ 9x^2 - 22x + 8 = 0 $.
Factor the quadratic expression.
Solve for $ x $ by setting each factor to zero.

Step 1: Rewrite the Equation

We start with the given equation: \[ 9x^2 = 22x - 8 \] Rearranging it to standard form gives: \[ 9x^2 - 22x + 8 = 0 \]

Step 2: Factor the Quadratic

Next, we factor the quadratic expression $ 9x^2 - 22x + 8 $. The factors of this expression are: \[ (9x - 2)(x - 4) = 0 \]

Step 3: Solve for $ x $

Using the zero-factor property, we set each factor equal to zero:

$ 9x - 2 = 0 $ leads to $ x = \frac{2}{9} $
$ x - 4 = 0 $ leads to $ x = 4 $

Final Answer

The solution set is: \[ \boxed{\left\{ \frac{2}{9}, 4 \right\}} \]

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