- Isolate the constant: \(x^2 + 5x = 10\).
- Add \((\frac{5}{2})^2 = \frac{25}{4}\) to both sides: \(x^2 + 5x + \frac{25}{4} = 10 + \frac{25}{4}\).
- Write as a perfect square and simplify: \((x + \frac{5}{2})^2 = \frac{65}{4}\).
- Solve for \(x\): \(x = \frac{-5 \pm \sqrt{65}}{2}\). The solution set is \(\boxed{\frac{-5 \pm \sqrt{65}}{2}}\).
Explanation
- Understanding the Problem
We are given the quadratic equation \(x^2 + 5x - 10 = 0\). Our goal is to solve for \(x\) using the method of completing the square. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\), where in this case \(a=1\), \(b=5\), and \(c=-10\). Completing the square involves rewriting the expression \(x^2 + bx\) into a perfect square trinomial \((x + h)^2\).
- Completing the Square Step 1
To begin, we isolate the variable terms by moving the constant term to the right side of the equation:
\[x^2 + 5x = 10\]
Next, we identify the coefficient of the \(x\)-term, which is \(b = 5\). To complete the square, we calculate \((\frac{b}{2})^2 = (\frac{5}{2})^2 = \frac{25}{4}\). We add this value to both sides of the equation to maintain equality:
\[x^2 + 5x + \frac{25}{4} = 10 + \frac{25}{4}\]
- Completing the Square Step 2
Now we rewrite the left side as a perfect square and simplify the right side:
\[(x + \frac{5}{2})^2 = \frac{40}{4} + \frac{25}{4} = \frac{65}{4}\]
Taking the square root of both sides gives us:
\[x + \frac{5}{2} = \pm \sqrt{\frac{65}{4}}\]
\[x + \frac{5}{2} = \pm \frac{\sqrt{65}}{2}\]
- Final Solution
Finally, we isolate \(x\) by subtracting \(\frac{5}{2}\) from both sides:
\[x = -\frac{5}{2} \pm \frac{\sqrt{65}}{2}\]
Thus, the solutions are:
\[x = \frac{-5 \pm \sqrt{65}}{2}\]
The solution set is \(\{\frac{-5 - \sqrt{65}}{2}, \frac{-5 + \sqrt{65}}{2}\}\).
Examples
Completing the square is a fundamental technique used in physics and engineering to analyze paths of projectiles. For example, if the height of a projectile is given by a quadratic function of time, you can complete the square to find the vertex of the parabola, which represents the maximum height and the time it occurs. This helps in understanding and predicting the behavior of moving objects.