¿Cuáles son las raíces del polinomio x²-5x+25?
Understand the Problem
La pregunta busca las raíces (o soluciones) del polinomio cuadrático dado, x²-5x+25. Esto implica encontrar los valores de 'x' que hacen que el polinomio sea igual a cero. Podemos usar la fórmula cuadrática para encontrar estas raíces.
Answer
$x = \frac{5}{2} \pm \frac{5\sqrt{3}}{2}i$
Answer for screen readers
$x = \frac{5}{2} + \frac{5\sqrt{3}}{2}i$ and $x = \frac{5}{2} - \frac{5\sqrt{3}}{2}i$
Steps to Solve
- Identify coefficients
The quadratic equation is given by $ax^2 + bx + c = 0$.
In the equation $x^2 - 5x + 25 = 0$, we have $a = 1$, $b = -5$, and $c = 25$.
- Apply the quadratic formula
The quadratic formula is:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
- Substitute the values
Substitute the values of $a$, $b$, and $c$ into the quadratic formula:
$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(25)}}{2(1)}$$
- Simplify the expression
Simplify the expression step by step:
$$x = \frac{5 \pm \sqrt{25 - 100}}{2}$$
$$x = \frac{5 \pm \sqrt{-75}}{2}$$
- Express the square root of a negative number using i
Since the discriminant is negative, we will have complex roots. Recall that $\sqrt{-1} = i$. $$\sqrt{-75} = \sqrt{75} \cdot \sqrt{-1} = \sqrt{25 \cdot 3} \cdot i = 5\sqrt{3}i$$
- Write the final solutions
Substitute this back into the equation for x:
$$x = \frac{5 \pm 5\sqrt{3}i}{2}$$
So, the solutions are:
$$x = \frac{5}{2} \pm \frac{5\sqrt{3}}{2}i$$
$x = \frac{5}{2} + \frac{5\sqrt{3}}{2}i$ and $x = \frac{5}{2} - \frac{5\sqrt{3}}{2}i$
More Information
The roots are complex conjugates. This is because the discriminant ($b^2 - 4ac$) was negative.
Tips
A common mistake is incorrectly substituting the values of $a$, $b$, and $c$ into the quadratic formula. Another common mistake is mishandling the negative sign inside the square root. Remember that $\sqrt{-1} = i$. It is also a common mistake to forget simplifying the radical, in our case $\sqrt{75}$
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