To solve the equation [tex]\(2x^2 + 3x - 7 = x^2 + 5x + 39\)[/tex], we need to rearrange and simplify it.
1. Move all the terms to one side of the equation:
Start by subtracting [tex]\(x^2 + 5x + 39\)[/tex] from both sides:
[tex]\[
2x^2 + 3x - 7 - (x^2 + 5x + 39) = 0
\][/tex]
2. Simplify the equation:
Combine like terms:
[tex]\[
2x^2 + 3x - 7 - x^2 - 5x - 39 = 0
\][/tex]
[tex]\[
(2x^2 - x^2) + (3x - 5x) + (-7 - 39) = 0
\][/tex]
[tex]\[
x^2 - 2x - 46 = 0
\][/tex]
3. Solve the quadratic equation [tex]\(x^2 - 2x - 46 = 0\)[/tex]:
A standard quadratic equation can be written in the form [tex]\(ax^2 + bx + c = 0\)[/tex]. In this equation, [tex]\(a = 1\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(c = -46\)[/tex].
We use the quadratic formula, which is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Plug [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the quadratic formula:
[tex]\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-46)}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{2 \pm \sqrt{4 + 184}}{2}
\][/tex]
[tex]\[
x = \frac{2 \pm \sqrt{188}}{2}
\][/tex]
[tex]\[
x = \frac{2 \pm \sqrt{4 \cdot 47}}{2}
\][/tex]
[tex]\[
x = \frac{2 \pm 2\sqrt{47}}{2}
\][/tex]
[tex]\[
x = 1 \pm \sqrt{47}
\][/tex]
4. Conclusion:
The solutions to the equation [tex]\(2x^2 + 3x - 7 = x^2 + 5x + 39\)[/tex] are:
[tex]\[
x = 1 - \sqrt{47} \quad \text{and} \quad x = 1 + \sqrt{47}
\][/tex]
Therefore, the answer is [tex]\( \boxed{1 \pm \sqrt{47}} \)[/tex].
