Complete the square so that the left side of x^2 + 8x - 3 = 0 becomes a binomial squared:
(x + __)^2 = __
The objective is to rewrite the quadratic equation \(x^2 + 8x - 3 = 0\) by completing the square to express it in the form \((x + k)^2 = m\). This process involves manipulating the original expression so that the terms involving \(x\) form a perfect square trinomial. We are given the original equation:
\[x^2 + 8x - 3 = 0\]
To start, we isolate the terms containing \(x\) on one side:
\[x^2 + 8x = 3\]
A perfect square trinomial follows the form \((x + k)^2 = x^2 + 2kx + k^2\). Comparing this to our expression \(x^2 + 8x\), we equate the coefficients of \(x\):
\[2k = 8\]
Solving for \(k\), we find:
\[k = \frac{8}{2} = 4\]
To complete the square, we need to add \(k^2\) to both sides of the equation. Calculating \(k^2\):
\[k^2 = 4^2 = 16\]
Now, we add \(16\) to both sides of the equation \(x^2 + 8x = 3\):
\[x^2 + 8x + 16 = 3 + 16\]
The left side is now a perfect square, \((x + 4)^2\). Simplifying the right side gives \(3 + 16 = 19\). Thus, the equation becomes:
\[(x + 4)^2 = 19\]
Comparing this to the required format \((x + \boxed{\phantom{0}})^2 = \boxed{\phantom{0}}\), we see that the first box is \(4\) and the second box is \(19\).
Completing the square is a powerful algebraic technique used to solve quadratic equations, find the vertex of a parabola, or simplify complex expressions. In real life, this is useful when modeling physical systems—for instance, calculating the trajectory of a projectile. If you were designing a fountain, you could use this method to determine the highest point the water arc reaches by identifying the vertex of its parabolic path, ensuring your design is both precise and aesthetically pleasing.