Find all solutions of the equation $$ x^{2}-8 x+36=0 $$ by using the quadratic formula. $$ x=\square $$

Question

UpStudy AI · Accepted Answer

Given the quadratic equation

$$
x^2 - 8x + 36 = 0,
$$

we identify the coefficients as $$a = 1$$, $$b = -8$$, and $$c = 36$$.

Using the quadratic formula

$$
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a},
$$

we substitute the coefficients:

$$
x = \frac{-(-8) \pm \sqrt{(-8)^2-4\cdot1\cdot36}}{2\cdot1}.
$$

Simplify step by step:

1. Calculate $$-(-8)$$:

$$
-(-8) = 8.
$$

2. Calculate the discriminant $$b^2 - 4ac$$:

$$
(-8)^2-4\cdot1\cdot36 = 64 - 144 = -80.
$$

3. Substitute the computed values back into the formula:

$$
x = \frac{8 \pm \sqrt{-80}}{2}.
$$

4. Simplify $$\sqrt{-80}$$ by expressing it in terms of $$i$$:

$$
\sqrt{-80} = \sqrt{80}\,i.
$$

5. Factor $$80$$ as $$16 \cdot 5$$:

$$
\sqrt{80} = \sqrt{16\cdot5} = \sqrt{16}\sqrt{5} = 4\sqrt{5}.
$$

Thus,

$$
\sqrt{-80} = 4\sqrt{5}\,i.
$$

6. Substitute back to get the solution:

$$
x = \frac{8 \pm 4\sqrt{5}\,i}{2} = 4 \pm 2\sqrt{5}\,i.
$$

So the solutions are:

$$
x = 4 + 2\sqrt{5}\,i \quad \text{and} \quad x = 4 - 2\sqrt{5}\,i.
$$
The solutions are $$ x = 4 + 2\sqrt{5}\,i $$ and $$ x = 4 - 2\sqrt{5}\,i $$.

Find all solutions of the equation \( x^{2}-8 x+36=0 \) by using the quadratic formula. \( x=\square \)