Solve by Completing the Square 1=x^2-6x
The problem presented requires the solution of a quadratic equation by the method of completing the square. The quadratic equation is in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Specifically, the given equation is \( 1 = x^2 - 6x \). To solve this equation by completing the square, one must manipulate the equation to express the left-hand side as a perfect square trinomial, which will allow the square root to be taken on both sides of the equation and thus isolating \( x \) to find the solutions. The completion of the square involves adding and subtracting the same value to preserve the original equation's integrity.
$1 = x^{2} - 6 x$
Answer
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 $\left(\frac{b}{2}\right)^2 = \left(-3\right)^2$.
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: $x = 6.16227766\ldots, -0.16227766\ldots$
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:
Ensuring the quadratic term has a coefficient of 1.
Moving the constant term to the opposite side of the equation.
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.
Adding this value to both sides of the equation.
Factoring the perfect square trinomial on one side.
Taking the square root of both sides to solve for $x$.
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 $\left(-3\right)^2 = 9$. 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.
