Problem 60 - Algebra A Combined Function

Question

Simplify. Assume that the variables represent any real number. $$ \sqrt{x^{2}-8 x+16} $$

Step-by-Step Solution

Answer

Simplified form is \(|x-4|\).

Step 1: Recognize the Expression Inside the Square Root

The expression inside the square root is \(x^2 - 8x + 16\). This is a quadratic expression.

Step 2: Factor the Quadratic Expression

Check if the quadratic expression \(x^2 - 8x + 16\) can be factored into a perfect square. Notice that \(x^2 - 8x + 16 = (x-4)^2\). This is because \((x-4)(x-4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16\).

Step 3: Simplify the Square Root

Since \(x^2 - 8x + 16 = (x-4)^2\), the square root is \(\sqrt{(x-4)^2}\). The square root of a square is the absolute value: \(|x-4|\).

Step 4: Consider the Values of the Expression

For the simplification \(|x-4|\), consider the two cases: If \(x \geq 4\), then \(|x-4| = x-4\). If \(x < 4\), then \(|x-4| = 4-x\). However, since the question allows all real numbers, the form \(|x-4|\) is correct and fully simplified.

Key Concepts

Quadratic Expressions

When dealing with expressions in algebra, quadratic expressions are of great importance. A quadratic expression is a polynomial of degree two, typically written in the form: \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, where \(a eq 0\). This kind of expression often represents a parabola when graphed on a coordinate plane.
In the original exercise, the expression inside the square root, \(x^2 - 8x + 16\), is a quadratic expression. It's important to recognize it as such, as this helps in simplifying and solving it effectively. Quadratic expressions can usually be factored or solved using various methods such as completing the square or using the quadratic formula.

Factoring Quadratics

Factoring is a key step when working with quadratic expressions, helping convert them into a product of simpler expressions. In our case, the quadratic \(x^2 - 8x + 16\) can be rewritten in a factored form. You can recognize it as a perfect square.

• Breaking it down: recognize that \(x^2 - 8x + 16\) equals \((x-4)^2\).
• Verification: Check by expanding \((x-4)(x-4)\) back to see \(x^2 - 8x + 16\). This confirms the factorization was accurate.

By factoring quadratics, you simplify the complexity of the expression, making subsequent operations such as simplification, solving, or graphing easier to manage.
Without this step, you may miss patterns that help solve the problem more efficiently.

Square Roots

Square roots are used frequently in algebra to simplify expressions and solve equations. The square root of a number is a value that, when multiplied by itself, yields the original number. For this reason, understanding the square roots of expressions, especially perfect squares, is crucial.
In the original exercise, we looked at the square root of \((x-4)^2\). Since \((x-4)^2\) is a perfect square, taking the square root simplifies directly to \(|x-4|\).
When simplifying square roots of perfect squares, it's important not to forget about absolute value, because the expression under the square root could have originally included negative numbers, affecting the result.

Absolute Value

Absolute value measures the distance of a number from zero on a number line, without considering which direction from zero the number lies.
It is always a non-negative value.
In the context of the original exercise, after simplifying the square root of \((x-4)^2\), it results in \(|x-4|\). This happens because when you take the square root of a squared term, the absolute value of the base term is returned.

• For \(x \geq 4\), the expression \(|x-4|\) simplifies to \(x-4\).
• For \(x < 4\), it becomes \(4-x\).

Absolute value ensures that expressions are simplified correctly across all potential negative domains, catering to the fact that all real numbers are considered in this simplification.