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Solve the equation 9, x, squared, plus, 17, x, plus, 9, equals, 3, x, squared9x
2
+17x+9=3x
2
to the nearest tenth.
Answer
Attempt 1 out of 5

Additional Solution
No Solution
x, equalsx=

Answer 1

To solve the equation \( 9x^2 + 17x + 9 = 3x^2 \), we first move all terms to one side of the equation to set it to zero:

\[
9x^2 + 17x + 9 - 3x^2 = 0
\]

This simplifies to:

\[
6x^2 + 17x + 9 = 0
\]

Next, we can use the quadratic formula to solve for \( x \). The quadratic formula is given by

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 6 \), \( b = 17 \), and \( c = 9 \).

Let's first calculate the discriminant \( b^2 - 4ac \):

\[
b^2 = 17^2 = 289
\]
\[
4ac = 4 \cdot 6 \cdot 9 = 216
\]
\[
b^2 - 4ac = 289 - 216 = 73
\]

Since the discriminant is positive, there will be two distinct real solutions. Now substituting back into the quadratic formula:

\[
x = \frac{-17 \pm \sqrt{73}}{2 \cdot 6}
\]
\[
x = \frac{-17 \pm \sqrt{73}}{12}
\]

Now, we need to calculate \( \sqrt{73} \):

\[
\sqrt{73} \approx 8.544
\]

Now substituting this back into our expression for \( x \):

1. For the plus case:
\[
x_1 = \frac{-17 + 8.544}{12} = \frac{-8.456}{12} \approx -0.7047
\]

2. For the minus case:
\[
x_2 = \frac{-17 - 8.544}{12} = \frac{-25.544}{12} \approx -2.1287
\]

Finally, rounding to the nearest tenth, we obtain:

\[
x_1 \approx -0.7
\]
\[
x_2 \approx -2.1
\]

Thus, the solutions to the equation \( 9x^2 + 17x + 9 = 3x^2 \) are:

\[
x \approx -0.7 \quad \text{and} \quad x \approx -2.1
\]