Solve $x^2-8 x=20$ by completing the square. Which is the solution set of the equation?
We are given the quadratic equation \(x^2 - 8x = 20\). Our goal is to solve this equation specifically by the method of completing the square. Completing the square is an algebraic technique used to rewrite a quadratic expression of the form \(x^2 + bx\) into a perfect square trinomial \((x+d)^2 - d^2\), which makes solving for \(x\) straightforward.
To complete the square for the expression \(x^2 - 8x\), we look at the coefficient of the \(x\) term, which is \(-8\). We divide this coefficient by 2 to get \(-4\), and then square this value to obtain \((-4)^2 = 16\). To maintain the equality, we must add this value to both sides of the original equation:
\[x^2 - 8x + 16 = 20 + 16\]
\[x^2 - 8x + 16 = 36\]
Now that we have a perfect square on the left-hand side, we can rewrite it as:
\[(x - 4)^2 = 36\]
To solve for \(x\), we take the square root of both sides, keeping in mind that the square root of \(36\) can be both positive and negative:
\[x - 4 = \pm \sqrt{36}\]
\[x - 4 = \pm 6\]
We now have two possible linear equations to solve:
1) \(x - 4 = 6 \implies x = 4 + 6 = 10\)
2) \(x - 4 = -6 \implies x = 4 - 6 = -2\)
Thus, the solutions to the equation are \(x = 10\) and \(x = -2\). The solution set is \(\{-2, 10\}\). Comparing this result with the given options, we select the option \(\{-2, 10\}\).
Completing the square is a powerful tool in engineering and physics, such as when determining the vertex of a parabolic trajectory. If you are calculating the height of a projectile given by the function \(h(t) = -t^2 + 4t + 10\), completing the square allows you to easily identify the time at which the projectile reaches its maximum height. Mastering this technique helps in modeling real-world motion, optimizing design parameters, and simplifying complex algebraic expressions.