Questions: Solve the quadratic equation for (x). [ (6 x-5)(x+9)=0 ] Enter the exact answers in increasing order. [ x=mathbfi x=square ]

Transcript text: Solve the quadratic equation for $x$. \[ (6 x-5)(x+9)=0 \] Enter the exact answers in increasing order. \[ \begin{array}{l} x=\mathbf{i} \\ x=\square \end{array} \]

Solution

Solution Steps

To solve the quadratic equation $(6x - 5)(x + 9) = 0$, we need to find the values of $x$ that make each factor equal to zero. This involves setting each factor equal to zero and solving for $x$.

Solution Approach

Set each factor equal to zero: $6x - 5 = 0$ and $x + 9 = 0$.
Solve each equation for $x$.
List the solutions in increasing order.

Step 1: Expand the Equation

We start with the given quadratic equation: \[ (6x - 5)(x + 9) = 0 \]

Step 2: Apply the Zero Product Property

The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero: \[ 6x - 5 = 0 \quad \text{or} \quad x + 9 = 0 \]

Step 3: Solve Each Equation

Solve the first equation for $ x $: \[ 6x - 5 = 0 \] Add 5 to both sides: \[ 6x = 5 \] Divide both sides by 6: \[ x = \frac{5}{6} \]

Solve the second equation for $ x $: \[ x + 9 = 0 \] Subtract 9 from both sides: \[ x = -9 \]

Step 4: Arrange the Solutions in Increasing Order

The solutions are $ x = -9 $ and $ x = \frac{5}{6} $. Arranging them in increasing order, we get: \[ x = -9, \quad x = \frac{5}{6} \]