Question

Solve the equation. Simplify all irrational solutions.

5x2 +14x = x + 6


(1 point)
Responses

x={−3 , 25}
x={−3 , 25}

x={−3, 5}
x= {−3, 5}

x={15, 5}

x={2/3, 3}

Answer 1

To solve the equation \(5x^2 + 14x = x + 6\), we first rearrange it into a standard form:

\[
5x^2 + 14x - x - 6 = 0
\]

This simplifies to:

\[
5x^2 + 13x - 6 = 0
\]

Now we can use the quadratic formula to find the solutions for \(x\). The quadratic formula is:

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

In our equation, \(a = 5\), \(b = 13\), and \(c = -6\). Plugging in these values:

1. Calculate the discriminant \(b^2 - 4ac\):

\[
b^2 - 4ac = 13^2 - 4(5)(-6) = 169 + 120 = 289
\]

2. Calculate \(x\):

\[
x = \frac{-13 \pm \sqrt{289}}{2 \cdot 5} = \frac{-13 \pm 17}{10}
\]

This gives us two potential solutions:

1.

\[
x = \frac{-13 + 17}{10} = \frac{4}{10} = \frac{2}{5}
\]

2.

\[
x = \frac{-13 - 17}{10} = \frac{-30}{10} = -3
\]

So the solutions are:

\[
x = -3 \quad \text{and} \quad x = \frac{2}{5}
\]

Now we compare against the provided response options:

- x={−3 , 25}
- x={−3, 5}
- x={15, 5}
- x={2/3, 3}

None of these options are correct based on our calculation of the solutions. The correct solutions are \(x = -3\) and \(x = \frac{2}{5}\). But the possible responses provided do not match our solutions.

If we are to also consider rounding or comparison of solutions, we conclude:

The simplest accepted answer is:

**\(x = \{-3, \frac{2}{5}\}\)** but not present in the options provided. Thus, none of the provided responses are correct.