Example Question - solutions
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Determining solutions of a quadratic equation using discriminant
Por supuesto, aunque no indicaste qué determinar exactamente de la ecuación cuadrática que se muestra: −2x^2 + 5x − 8 = 0 Generalmente, "determinar, sin resolver la ecuación" podría significar identificar el número y naturaleza de las soluciones sin calcular los valores exactos. Para ello, podemos usar el discriminante de una ecuación cuadrática, el cual se define como D = b^2 - 4ac, donde a, b y c son los coeficientes de x^2, x y el término constante, respectivamente. En este caso, los coeficientes son: a = -2 b = 5 c = -8 Calculamos el discriminante: D = b^2 - 4ac D = (5)^2 - 4(-2)(-8) D = 25 - 4(16) D = 25 - 64 D = -39 Dado que el discriminante es negativo (D < 0), podemos determinar que la ecuación tiene dos soluciones complejas conjugadas. Esto significa que no hay soluciones reales para la ecuación cuadrática dada.
Quadratic Formula and Coefficients
The equation in the image appears to be the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] For a quadratic equation in the form \(ax^2 + bx + c = 0\). However, it appears that there is a missing term in the denominator (it should be \(2a\), but only \(2(1)\) is written), and the coefficients 'a', 'b', and 'c' are not specified in the image. The formula shown is: \[x = \frac{-1 \pm \sqrt{1^2 - 4(8)(20)}}{2(1)}\] This suggests 'a' = 8, 'b' = 1, and 'c' = 20 for the quadratic equation. However, given the typical structure of a quadratic equation \(ax^2 + bx + c = 0\), it's unusual to have an 'a' term larger than 'b' or 'c'. It's possible there's a mistake in the handwriting. Assuming the 'a', 'b', and 'c' values as 1, 1, and -160 respectively (if we take the coefficient of 'a' to be 1 instead of 8), the equation would be: \[x = \frac{-1 \pm \sqrt{(1)^2 - 4(1)(-160)}}{2(1)}\] Now let's solve it with these coefficients: \[x = \frac{-1 \pm \sqrt{1 + 640}}{2}\] \[x = \frac{-1 \pm \sqrt{641}}{2}\] We now have two possible solutions for x, which are: \[x_1 = \frac{-1 + \sqrt{641}}{2}\] \[x_2 = \frac{-1 - \sqrt{641}}{2}\] These are the two solutions to the quadratic equation based on the given formula and assuming 'a' = 1. If 'a' is actually 8 as shown, then the quadratic equation would be incorrect, and we would need the correct coefficients to proceed.
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