¿Cuáles son las raíces del polinomio x² - 5x + 25?
Understand the Problem
La pregunta busca las raíces (o ceros) del polinomio cuadrático x² - 5x + 25. Esto implica encontrar los valores de 'x' que hacen que el polinomio sea igual a cero. Podemos utilizar la fórmula cuadrática para encontrar estas raíces.
Answer
$x = \frac{5}{2} \pm \frac{5\sqrt{3}}{2}i$
Answer for screen readers
$x = \frac{5}{2} + \frac{5\sqrt{3}}{2}i$, $x = \frac{5}{2} - \frac{5\sqrt{3}}{2}i$
Steps to Solve
- Identify the coefficients
First, identify the coefficients $a$, $b$, and $c$ in the quadratic equation $ax^2 + bx + c = 0$. In this case, the equation is $x^2 - 5x + 25 = 0$, so $a = 1$, $b = -5$, and $c = 25$.
- Apply the quadratic formula
The quadratic formula is given by:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Substitute the values of $a$, $b$, and $c$ into the formula:
$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(25)}}{2(1)}$$
- Simplify the expression
Simplify the expression step by step:
$$x = \frac{5 \pm \sqrt{25 - 100}}{2}$$
$$x = \frac{5 \pm \sqrt{-75}}{2}$$
- Handle the negative square root
Since the value inside the square root is negative, we will have complex roots. We can rewrite $\sqrt{-75}$ as $\sqrt{75}i$, where $i$ is the imaginary unit ($i = \sqrt{-1}$). Furthermore, simplify $\sqrt{75}$ as $\sqrt{25 \cdot 3} = 5\sqrt{3}$. Thus, $\sqrt{-75} = 5\sqrt{3}i$:
$$x = \frac{5 \pm 5\sqrt{3}i}{2}$$
- Express the roots in simplest form
The roots are:
$$x = \frac{5}{2} \pm \frac{5\sqrt{3}}{2}i$$
$x = \frac{5}{2} + \frac{5\sqrt{3}}{2}i$, $x = \frac{5}{2} - \frac{5\sqrt{3}}{2}i$
More Information
The roots are complex conjugates of each other. This is a hallmark of quadratic equations with real coefficients and a negative discriminant.
Tips
A common mistake is incorrectly applying the quadratic formula, especially with the signs. Another mistake is incorrectly simplifying the square root of a negative number. Remember that $\sqrt{-1} = i$.
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