Find the Roots (Zeros) x-6 = square root of x

Solution:

Step 1:

Move the radical to the left side by reversing the equation: $x - 6 = \sqrt{x}$ becomes $\sqrt{x} = x - 6$.

Step 2:

Eliminate the square root by squaring both sides: $(\sqrt{x})^2 = (x - 6)^2$.

Step 3:

Simplify the equation.

Step 3.1:

Express $\sqrt{x}$ as $x^{1/2}$: $(x^{1/2})^2 = (x - 6)^2$.

Step 3.2:

Simplify the left-hand side.

Step 3.2.1:

Square $x^{1/2}$.

Step 3.2.1.1:

Apply the exponent rule: $(a^m)^n = a^{mn}$.

Step 3.2.1.1.1:

Multiply the exponents: $x^{1/2 \cdot 2} = (x - 6)^2$.

Step 3.2.1.1.2:

Simplify the exponent: $x^{1} = (x - 6)^2$.

Step 3.2.1.2:

Resulting in $x = (x - 6)^2$.

Step 3.3:

Simplify the right-hand side.

Step 3.3.1:

Expand $(x - 6)^2$.

Step 3.3.1.1:

Express as a product: $x = (x - 6)(x - 6)$.

Step 3.3.1.2:

Apply the FOIL method to multiply.

Step 3.3.1.2.1:

Distribute: $x = x(x - 6) - 6(x - 6)$.

Step 3.3.1.2.2:

Continue distribution: $x = x^2 - 6x - 6x + 36$.

Step 3.3.1.3:

Combine like terms: $x = x^2 - 12x + 36$.

Step 4:

Isolate $x$ to solve the equation.

Step 4.1:

Rearrange: $x^2 - 12x + 36 = x$.

Step 4.2:

Move all $x$ terms to one side: $x^2 - 12x + 36 - x = 0$.

Step 4.2.1:

Combine $x$ terms: $x^2 - 13x + 36 = 0$.

Step 4.3:

Factor the quadratic equation: $x^2 - 13x + 36$.

Step 4.3.1:

Find two numbers that multiply to $36$ and add to $-13$: $-9$ and $-4$.

Step 4.3.2:

Write the factors: $(x - 9)(x - 4) = 0$.

Step 4.4:

Set each factor equal to zero: $x - 9 = 0$ and $x - 4 = 0$.

Step 4.5:

Solve for $x$ from $x - 9 = 0$: $x = 9$.

Step 4.6:

Solve for $x$ from $x - 4 = 0$: $x = 4$.

Step 4.7:

The potential solutions are $x = 9$ and $x = 4$.

Step 5:

Check which solutions satisfy the original equation: $x = 9$ is the only valid solution.

Step 6:

The final answer is $x = 9$.

Knowledge Notes:

1. Square Roots and Exponents: The square root of a number $x$ is written as $\sqrt{x}$ and is equivalent to $x^{1/2}$. Squaring a square root, $(\sqrt{x})^2$, eliminates the radical, leaving $x$.

2. Rearranging Equations: Equations can be rearranged to move terms from one side to another, often to isolate the variable we are solving for.

3. Squaring Both Sides: When an equation contains a square root, we can square both sides to remove the radical. This must be done carefully, as it can introduce extraneous solutions.

4. FOIL Method: This is a technique for expanding the product of two binomials. It stands for First, Outer, Inner, Last, referring to the terms in each binomial that are multiplied together.

5. Factoring Quadratics: The process of breaking down a quadratic equation into the product of two binomials. The AC method involves finding two numbers that multiply to the constant term and add to the coefficient of the linear term.

6. Zero Product Property: If the product of two factors is zero, then at least one of the factors must be zero. This property is used to solve quadratic equations that have been factored into binomials.

7. Checking Solutions: It's important to substitute the solutions back into the original equation to ensure they are valid, as squaring both sides can introduce solutions that do not actually satisfy the original equation.