Solve by Completing the Square 1=x^2-6x

Solution:

Step 1:

Move the $x$ terms to the left by reversing the equation: $x^2-6x=1$.

Step 2:

Determine the number to complete the square, which is ${\left(\frac{b}{2}\right)}^2={(-3)}^2$.

Step 3:

Add this number to both sides of the equation: $x^2-6x+(-3{)}^2=1+(-3{)}^2$.

Step 4:

Simplify both sides of the equation.

Step 4.1:

On the left, combine like terms.

Step 4.1.1:

Calculate $(-3{)}^2$: $x^2-6x+9=1+(-3{)}^2$.

Step 4.2:

On the right, simplify the expression.

Step 4.2.1:

Evaluate $1+(-3{)}^2$.

Step 4.2.1.1:

Square $-3$: $x^2-6x+9=1+9$.

Step 4.2.1.2:

Add $1$ and $9$: $x^2-6x+9=10$.

Step 5:

Factor the left side as a binomial square: $(x-3{)}^2=10$.

Step 6:

Solve for $x$.

Step 6.1:

Take the square root of both sides: $x-3=\pm \sqrt{10}$.

Step 6.2:

Isolate $x$ by adding $3$ to both sides: $x=\pm \sqrt{10}+3$.

Step 7:

Present the solution in its exact and decimal forms.

Exact Form: $x=\pm \sqrt{10}+3$ Decimal Form: $x=6.16227766\dots ,-0.16227766\dots$

Knowledge Notes:

To solve a quadratic equation by completing the square, one must manipulate the equation to form a perfect square trinomial on one side. The steps typically involve:

1. Ensuring the quadratic term has a coefficient of 1.

2. Moving the constant term to the opposite side of the equation.

3. Finding the value needed to complete the square, which is ${\left(\frac{b}{2}\right)}^2$, where $b$ is the coefficient of the $x$ term.

4. Adding this value to both sides of the equation.

5. Factoring the perfect square trinomial on one side.

6. Taking the square root of both sides to solve for $x$.

7. Isolating $x$ and simplifying as needed.

In this case, the quadratic equation is $x^2-6x=1$. The value needed to complete the square is ${(-3)}^2=9$. After adding 9 to both sides and factoring, we get $(x-3{)}^2=10$. Taking the square root gives $x-3=\pm \sqrt{10}$, and adding 3 to both sides gives the solutions $x=\pm \sqrt{10}+3$. The decimal form of the solutions can be found using a calculator.