(2x-3)°+(3+4x)•(x+5)=82

To solve the equation \((2x - 3)^\circ + (3 + 4x) \cdot (x + 5) = 82\), we'll first clarify the components. We interpret \((2x - 3)^\circ\) as a temperature in degrees, but in a mathematical context, we can treat it as a linear expression. Here’s how to solve it step by step:

1. Expand the equation: \[ (3 + 4x)(x + 5) = 3x + 15 + 4x^2 + 20x = 4x^2 + 23x + 15 \]

2. Rewrite the equation: \[ (2x - 3) + (4x^2 + 23x + 15) = 82 \]

3. Combine like terms: \[ 4x^2 + (23x + 2x) + (15 - 3) = 82 \] \[ 4x^2 + 25x + 12 = 82 \]

4. Move all terms to one side: \[ 4x^2 + 25x + 12 - 82 = 0 \] \[ 4x^2 + 25x - 70 = 0 \]

5. Now, use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 4\), \(b = 25\), and \(c = -70\).

6. Calculate the discriminant: \[ b^2 - 4ac = 25^2 - 4 \cdot 4 \cdot (-70) = 625 + 1120 = 1745 \]

7. Solve for \(x\): \[ x = \frac{-25 \pm \sqrt{1745}}{8} \]

8. Calculate \(\sqrt{1745}\): \(\sqrt{1745} \approx 41.8\)

   So, \[ x \approx \frac{-25 \pm 41.8}{8} \]

9. Calculate the two potential solutions:

   • For \(x_1\): \[ x_1 = \frac{-25 + 41.8}{8} \approx \frac{16.8}{8} \approx 2.1 \]
   • For \(x_2\): \[ x_2 = \frac{-25 - 41.8}{8} \approx \frac{-66.8}{8} \approx -8.35 \]

The solutions to the equation \((2x - 3) + (3 + 4x)(x + 5) = 82\) are approximately:

\[ x \approx 2.1 \quad \text{and} \quad x \approx -8.35 \]