Solve the equation for all real solutions in simplest form. $$ x^{2}-7 x+2=0 $$

Question

UpStudy AI · Accepted Answer

To solve the quadratic equation:

$$
x^{2} - 7x + 2 = 0
$$

we can use the **quadratic formula**, which is given by:

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

For the equation $$ ax^2 + bx + c = 0 $$:

- $$ a = 1 $$
- $$ b = -7 $$
- $$ c = 2 $$

Plugging these values into the quadratic formula:

$$
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(2)}}{2(1)} = \frac{7 \pm \sqrt{49 - 8}}{2} = \frac{7 \pm \sqrt{41}}{2}
$$

So, the solutions are:

$$
x = \frac{7 + \sqrt{41}}{2} \quad \text{and} \quad x = \frac{7 - \sqrt{41}}{2}
$$

**Final Answer:**

All real solutions are (7 ± √41) divided by two. In symbols,

x = ( 7 ± √41 ) ⁄ 2
The solutions are $$ x = \frac{7 + \sqrt{41}}{2} $$ and $$ x = \frac{7 - \sqrt{41}}{2} $$.

Solve the equation for all real solutions in simplest form. \[ x^{2}-7 x+2=0 \]