Solve each equation. (x-4)^2/5 = 9

Verified step by step guidance

1. Start by isolating the squared term. Multiply both sides of the equation by 5 to eliminate the fraction: $(x-4{)}^2=45$.
2. Take the square root of both sides to solve for $x-4$. Remember to consider both the positive and negative roots: $x-4=\pm \sqrt{45}$.
3. Simplify $\sqrt{45}$ by expressing it as $\sqrt{9\times 5}=3\sqrt{5}$.
4. Set up two separate equations to solve for $x$: $x-4=3\sqrt{5}$ and $x-4=-3\sqrt{5}$.
5. Solve each equation for $x$ by adding 4 to both sides: $x=4+3\sqrt{5}$ and $x=4-3\sqrt{5}$.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. They can be solved using various methods, including factoring, completing the square, or applying the quadratic formula. In this problem, the equation involves a squared term, indicating that it can be treated as a quadratic equation once simplified.

Square Roots

Square roots are the values that, when multiplied by themselves, yield the original number. In solving equations involving squares, such as (x-4)^2, taking the square root is a crucial step. This process introduces both positive and negative solutions, which must be considered to find all possible values of x.

Isolating Variables

Isolating variables is a fundamental algebraic technique used to solve equations. It involves rearranging the equation to get the variable of interest on one side and all other terms on the opposite side. In this case, isolating x requires manipulating the equation to eliminate the fraction and the square, allowing for straightforward calculation of the variable's value.