¿Cuáles son las raíces del polinomio x^2 - 5x + 25?
Understand the Problem
La pregunta es sobre encontrar las raíces de un polinomio cuadrático. Necesitamos aplicar la fórmula cuadrática o completar el cuadrado para hallar las soluciones (raíces) de la ecuación dada.
Answer
$x = \frac{5}{2} \pm \frac{5\sqrt{3}}{2} i$
Answer for screen readers
b) (ver imagen a pantalla completa) $x = \frac{5}{2} \pm \frac{5\sqrt{3}}{2} i$
Steps to Solve
- Identify the coefficients
The quadratic equation is given by $x^2 - 5x + 25 = 0$. Here, $a = 1$, $b = -5$, and $c = 25$.
- Apply the quadratic formula
The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Substituting the values of $a$, $b$, and $c$, we get: $$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(25)}}{2(1)}$$
- Simplify the expression
$$x = \frac{5 \pm \sqrt{25 - 100}}{2}$$ $$x = \frac{5 \pm \sqrt{-75}}{2}$$
- Simplify the square root
Since we have a negative number under the square root, we will have complex roots. We can rewrite $\sqrt{-75}$ as $\sqrt{75}i$. $$\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}$$ So, $\sqrt{-75} = 5\sqrt{3}i$.
- Write the roots
Substituting back into the equation for $x$: $$x = \frac{5 \pm 5\sqrt{3}i}{2}$$ $$x = \frac{5}{2} \pm \frac{5\sqrt{3}}{2}i$$
b) (ver imagen a pantalla completa) $x = \frac{5}{2} \pm \frac{5\sqrt{3}}{2} i$
More Information
The roots are complex conjugates of each other. This is expected since the discriminant ($b^2 - 4ac$) is negative.
Tips
A common mistake is to miscalculate the discriminant ($b^2 - 4ac$) or to incorrectly simplify the square root. Another common mistake is to forget the $\pm$ sign in the quadratic formula, or to not realize that the square root of a negative number results in an imaginary number.
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