- Set the quadratic function equal to zero: \(2x^2 + 8x - 3 = 0\).
- Factor out the leading coefficient and add the value required to complete the square: \(2(x + 2)^2 = 11\).
- Solve for \(x\) by isolating the squared binomial and taking the square root.
- The zeros are \(\boxed{x = -2 \pm \sqrt{\frac{11}{2}}}\).
Explanation
- Understanding the Problem and Objective
To find the zeros of the quadratic function \(f(x) = 2x^2 + 8x - 3\), we must solve the equation \(f(x) = 0\). The goal is to set \(2x^2 + 8x - 3 = 0\) and use the completing the square method to isolate \(x\). This method involves rearranging the quadratic into the form \(a(x + h)^2 + k = 0\), which allows us to solve for \(x\) directly by taking square roots.
- Setting up and Factoring the Quadratic Equation
First, we set the function equal to zero:
\[2x^2 + 8x - 3 = 0.\]
Next, isolate the constant term by adding \(3\) to both sides:
\[2x^2 + 8x = 3.\]
Now, factor out the leading coefficient, \(2\), from the \(x\)-terms to simplify the expression for completing the square:
\[2(x^2 + 4x) = 3.\]
- Completing the Square
To complete the square for the expression \(x^2 + 4x\), we take half of the coefficient of \(x\), which is \(4\), divide it by \(2\), and then square the result:
\[\left(\frac{4}{2}\right)^2 = 2^2 = 4.\]
Now, add this value inside the parentheses. Since we have a multiplier of \(2\) on the outside, we must add \(2 \times 4 = 8\) to the right side of the equation to keep it balanced:
\[2(x^2 + 4x + 4) = 3 + 8.\]
Simplifying the right side gives:
\[2(x^2 + 4x + 4) = 11.\]
- Solving for the Zeros
The expression inside the parentheses, \(x^2 + 4x + 4\), is a perfect square binomial, \((x + 2)^2\). We can rewrite the equation as:
\[2(x + 2)^2 = 11.\]
Divide both sides by \(2\) to isolate the squared binomial:
\[(x + 2)^2 = \frac{11}{2}.\]
Now, take the square root of both sides:
\[x + 2 = \pm \sqrt{\frac{11}{2}}.\]
Finally, solve for \(x\) by subtracting \(2\) from both sides:
\[x = -2 \pm \sqrt{\frac{11}{2}}.\]
Thus, the zeros of the function are \(x = -2 - \sqrt{\frac{11}{2}}\) and \(x = -2 + \sqrt{\frac{11}{2}}\). This corresponds to the first option provided.
Examples
Completing the square is a powerful tool in physics and engineering. For example, if you are modeling the trajectory of a projectile launched at an angle, the vertical position over time can be represented by a quadratic function. By completing the square, you can quickly identify the vertex of the parabola, which tells you the maximum height reached and at what time it occurs. This helps in predicting the peak performance or limitations of mechanical systems and athletic maneuvers alike.