Question

to solve 9x^2-12x+4=49 by using the square root property you would first rewrite the equation as acter doign the factor/sum

Answer 1

To solve the equation \( 9x^2 - 12x + 4 = 49 \) using the square root property, we'll follow these steps:

1. **Rearrange the equation**: First, we need to bring all terms to one side of the equation so that one side equals zero.

\[
9x^2 - 12x + 4 - 49 = 0
\]
\[
9x^2 - 12x - 45 = 0
\]

2. **Factor the quadratic**: Next, we look to factor the left-hand side. First, we can simplify the quadratic equation by dividing everything by 3:

\[
3x^2 - 4x - 15 = 0
\]

Now, we'll factor this quadratic. We need to find two numbers that multiply to \( 3 \cdot (-15) = -45 \) and add to \(-4\). Those numbers are \(-9\) and \(5\).

We can rewrite the middle term:

\[
3x^2 - 9x + 5x - 15 = 0
\]

Grouping the terms, we get:

\[
3x(x - 3) + 5(x - 3) = 0
\]
\[
(3x + 5)(x - 3) = 0
\]

3. **Set each factor to zero**: Now we set each factor equal to zero to find the values of \(x\).

\[
3x + 5 = 0 \quad \text{or} \quad x - 3 = 0
\]

For \(3x + 5 = 0\):

\[
3x = -5 \quad \Rightarrow \quad x = -\frac{5}{3}
\]

For \(x - 3 = 0\):

\[
x = 3
\]

4. **Conclusion**: Thus, the solutions to the equation \(9x^2 - 12x + 4 = 49\) are

\[
x = -\frac{5}{3} \quad \text{and} \quad x = 3.
\]

If you'd like, we could also use the square root property directly after completing the square. Would you like to see that method as well?