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Matemáticas 3 · Tema 8

Fórmula general

Ejemplo

x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
−5x2+3x+2=0-5x^2 + 3x + 2 = 0
x=−(3)±(3)2−4(−5)(2)2(−5)x=−3±9+40−10x=−3±49−10x=−3±7−10\begin{aligned} x &= \dfrac{-(3) \pm \sqrt{(3)^2 - 4(-5)(2)}}{2(-5)} \\ x &= \dfrac{-3 \pm \sqrt{9 + 40}}{-10} \\ x &= \dfrac{-3 \pm \sqrt{49}}{-10} \\ x &= \dfrac{-3 \pm 7}{-10} \end{aligned}
x1=−3+7−10x1=4−10\begin{aligned} x_1 &= \dfrac{-3 + 7}{-10} \\ x_1 &= \dfrac{4}{-10} \end{aligned} x1=−0.4x_1 = -0.4
x2=−3−7−10x2=−10−10\begin{aligned} x_2 &= \dfrac{-3 - 7}{-10} \\ x_2 &= \dfrac{-10}{-10} \end{aligned} x2=1x_2 = 1

Ejercicios

Ejercicio 1
3x2−4x−1=03x^2 - 4x - 1 = 0
Ejercicio 2
−2x2+3x−5=0-2x^2 + 3x - 5 = 0
Ejercicio 3
2x2+6x−3=02x^2 + 6x - 3 = 0
Ejercicio 4
−5x2+4x+2=0-5x^2 + 4x + 2 = 0
Ejercicio 5
3x2−2x−1=03x^2 - 2x - 1 = 0

Respuestas

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Ejercicio 1
3x2−4x−1=03x^2 - 4x - 1 = 0
x=−(−4)±(−4)2−4(3)(−1)2(3)x=4±16+126x=4±286x≈4±5.36\begin{aligned} x &= \dfrac{-(-4) \pm \sqrt{(-4)^2 - 4(3)(-1)}}{2(3)} \\ x &= \dfrac{4 \pm \sqrt{16 + 12}}{6} \\ x &= \dfrac{4 \pm \sqrt{28}}{6} \\ x &\approx \dfrac{4 \pm 5.3}{6} \end{aligned}
x1≈4+5.36x1≈9.36\begin{aligned} x_1 &\approx \dfrac{4 + 5.3}{6} \\ x_1 &\approx \dfrac{9.3}{6} \end{aligned} x1≈1.55x_1 \approx 1.55
x2≈4−5.36x2≈−1.36\begin{aligned} x_2 &\approx \dfrac{4 - 5.3}{6} \\ x_2 &\approx \dfrac{-1.3}{6} \end{aligned} x2≈−0.21x_2 \approx -0.21
Ejercicio 2
−2x2+3x−5=0-2x^2 + 3x - 5 = 0
x=−(3)±(3)2−4(−2)(−5)2(−2)x=−3±9−40−4x=−3±−31−4\begin{aligned} x &= \dfrac{-(3) \pm \sqrt{(3)^2 - 4(-2)(-5)}}{2(-2)} \\ x &= \dfrac{-3 \pm \sqrt{9 - 40}}{-4} \\ x &= \dfrac{-3 \pm \sqrt{-31}}{-4} \end{aligned}

La raíz da negativo: no existe solución en R\mathbb{R}.

Ejercicio 3
2x2+6x−3=02x^2 + 6x - 3 = 0
x=−(6)±(6)2−4(2)(−3)2(2)x=−6±36+244x=−6±604x≈−6±7.74\begin{aligned} x &= \dfrac{-(6) \pm \sqrt{(6)^2 - 4(2)(-3)}}{2(2)} \\ x &= \dfrac{-6 \pm \sqrt{36 + 24}}{4} \\ x &= \dfrac{-6 \pm \sqrt{60}}{4} \\ x &\approx \dfrac{-6 \pm 7.7}{4} \end{aligned}
x1≈−6+7.74x1≈1.74\begin{aligned} x_1 &\approx \dfrac{-6 + 7.7}{4} \\ x_1 &\approx \dfrac{1.7}{4} \end{aligned} x1≈0.42x_1 \approx 0.42
x2≈−6−7.74x2≈−13.74\begin{aligned} x_2 &\approx \dfrac{-6 - 7.7}{4} \\ x_2 &\approx \dfrac{-13.7}{4} \end{aligned} x2≈−3.42x_2 \approx -3.42
Ejercicio 4
−5x2+4x+2=0-5x^2 + 4x + 2 = 0
x=−(4)±(4)2−4(−5)(2)2(−5)x=−4±16+40−10x=−4±56−10x≈−4±7.5−10\begin{aligned} x &= \dfrac{-(4) \pm \sqrt{(4)^2 - 4(-5)(2)}}{2(-5)} \\ x &= \dfrac{-4 \pm \sqrt{16 + 40}}{-10} \\ x &= \dfrac{-4 \pm \sqrt{56}}{-10} \\ x &\approx \dfrac{-4 \pm 7.5}{-10} \end{aligned}
x1≈−4+7.5−10x1≈3.5−10\begin{aligned} x_1 &\approx \dfrac{-4 + 7.5}{-10} \\ x_1 &\approx \dfrac{3.5}{-10} \end{aligned} x1≈−0.35x_1 \approx -0.35
x2≈−4−7.5−10x2≈−11.5−10\begin{aligned} x_2 &\approx \dfrac{-4 - 7.5}{-10} \\ x_2 &\approx \dfrac{-11.5}{-10} \end{aligned} x2≈1.15x_2 \approx 1.15
Ejercicio 5
3x2−2x−1=03x^2 - 2x - 1 = 0
x=−(−2)±(−2)2−4(3)(−1)2(3)x=2±4+126x=2±166x=2±46\begin{aligned} x &= \dfrac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(-1)}}{2(3)} \\ x &= \dfrac{2 \pm \sqrt{4 + 12}}{6} \\ x &= \dfrac{2 \pm \sqrt{16}}{6} \\ x &= \dfrac{2 \pm 4}{6} \end{aligned}
x1=2+46x1=66\begin{aligned} x_1 &= \dfrac{2 + 4}{6} \\ x_1 &= \dfrac{6}{6} \end{aligned} x1=1x_1 = 1
x2=2−46x2=−26\begin{aligned} x_2 &= \dfrac{2 - 4}{6} \\ x_2 &= \dfrac{-2}{6} \end{aligned} x2≈−0.3x_2 \approx -0.3

Ejercicios extra

▼
Bloque 1
x2−5x+6=0x^2 - 5x + 6 = 0
2x2+3x−2=02x^2 + 3x - 2 = 0
x2+2x−8=0x^2 + 2x - 8 = 0
3x2+5x+1=03x^2 + 5x + 1 = 0
−x2+4x+3=0-x^2 + 4x + 3 = 0
Bloque 2
2x2−7x+3=02x^2 - 7x + 3 = 0
4x2+4x+1=04x^2 + 4x + 1 = 0
x2−3x+5=0x^2 - 3x + 5 = 0
5x2−2x−4=05x^2 - 2x - 4 = 0
−3x2+6x+2=0-3x^2 + 6x + 2 = 0
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