To solve the equation [tex]\((x + 2)(x - 7) = 3\)[/tex], let's work through it step by step:
1. Expand the Left-Hand Side: First, let's expand the left-hand side of the equation:
[tex]\[
(x + 2)(x - 7) = x^2 - 7x + 2x - 14 = x^2 - 5x - 14
\][/tex]
So, the equation becomes:
[tex]\[
x^2 - 5x - 14 = 3
\][/tex]
2. Move All Terms to One Side: Next, we move all terms to one side to set the equation to zero:
[tex]\[
x^2 - 5x - 14 - 3 = 0
\][/tex]
Simplifying this, we have:
[tex]\[
x^2 - 5x - 17 = 0
\][/tex]
3. Solve the Quadratic Equation: The next step is to solve this quadratic equation [tex]\(x^2 - 5x - 17 = 0\)[/tex] using the quadratic formula, which is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(c = -17\)[/tex]. Plugging these values into the formula, we get:
[tex]\[
x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-17)}}{2 \cdot 1}
\][/tex]
Simplifying inside the square root:
[tex]\[
x = \frac{5 \pm \sqrt{25 + 68}}{2} = \frac{5 \pm \sqrt{93}}{2}
\][/tex]
4. Split into Two Solutions: This gives us two solutions:
[tex]\[
x = \frac{5 - \sqrt{93}}{2} \quad \text{or} \quad x = \frac{5 + \sqrt{93}}{2}
\][/tex]
These are the solutions to the equation:
[tex]\[
x = \frac{5 - \sqrt{93}}{2} \quad \text{and} \quad x = \frac{5 + \sqrt{93}}{2}
\][/tex]
Therefore, the solutions to the equation [tex]\((x + 2)(x - 7) = 3\)[/tex] are:
[tex]\[
x = \frac{5}{2} - \frac{\sqrt{93}}{2} \quad \text{and} \quad x = \frac{5}{2} + \frac{\sqrt{93}}{2}
\][/tex]
