Questions: Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying (x)-intercepts. [ x^2-3 x-54=0 ] The solution set is (square).

Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying (x)-intercepts.
[ x^2-3 x-54=0 ]

The solution set is (square).

Transcript text: Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying $x$-intercepts. \[ x^{2}-3 x-54=0 \] The solution set is $\square$.

Solution

Solution Steps

To solve the quadratic equation $x^2 - 3x - 54 = 0$ by factoring, we need to find two numbers that multiply to -54 and add up to -3. Once we find these numbers, we can factor the quadratic equation and solve for $x$. Finally, we can verify the solutions by substitution.

Step 1: Factor the Quadratic Equation

To solve the quadratic equation $x^2 - 3x - 54 = 0$ by factoring, we need to find two numbers that multiply to $-54$ and add up to $-3$. These numbers are $-9$ and $6$.

Step 2: Write the Factored Form

Using the numbers $-9$ and $6$, we can write the factored form of the quadratic equation: \[ x^2 - 3x - 54 = (x - 9)(x + 6) = 0 \]

Step 3: Solve for $x$

Set each factor equal to zero and solve for $x$: \[ x - 9 = 0 \quad \Rightarrow \quad x = 9 \] \[ x + 6 = 0 \quad \Rightarrow \quad x = -6 \]

Step 4: Verify the Solutions

Substitute $x = 9$ and $x = -6$ back into the original equation to verify: \[ 9^2 - 3(9) - 54 = 81 - 27 - 54 = 0 \] \[ (-6)^2 - 3(-6) - 54 = 36 + 18 - 54 = 0 \] Both solutions satisfy the original equation.

Final Answer

The solution set is: \[ \boxed{x = -6, 9} \]

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