Question

solve the quadratic equation by completing the square.
$x^{2}-18x + 78 = 0$
first, choose the appropriate form and fill in the blanks with the correct numbers.
then, solve the equation. simplify your answer as much as possible.
if there is more than one solution, separate them with commas.
form:
$\circ(x + \square)^{2}=\square$
$\circ(x - \square)^{2}=\square$
solution:
$x = \square$

Explanation:

Step1: Rearrange the equation

Given $x^{2}-18x + 78=0$, we can rewrite it as $x^{2}-18x=-78$.

Step2: Complete the square

For the quadratic expression $x^{2}-18x$, the coefficient of $x$ is $- 18$. Half of it is $\frac{-18}{2}=-9$, and its square is $(-9)^{2} = 81$. Add 81 to both sides of the equation: $x^{2}-18x + 81=-78 + 81$.

Step3: Factor the left - hand side

The left - hand side can be factored as $(x - 9)^{2}$, and the right - hand side is 3. So, $(x - 9)^{2}=3$.

Step4: Solve for x

Take the square root of both sides: $x-9=\pm\sqrt{3}$. Then $x = 9\pm\sqrt{3}$.

Answer:

Form: $(x - 9)^{2}=3$
Solution: $x=9+\sqrt{3},9 - \sqrt{3}$