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Matemáticas 3 · Tema 7

Ecuaciones cuadráticas incompletas

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Ejemplo

Puras
ax2+c=0ax^2 + c = 0 4x2−5=7x2−104x2−7x2=−10+5−3x2=−5x2=−5−3=53x2=±53\begin{aligned} 4x^2 - 5 &= 7x^2 - 10 \\ 4x^2 - 7x^2 &= -10 + 5 \\ -3x^2 &= -5 \\ x^2 &= \dfrac{-5}{-3} = \dfrac{5}{3} \\ \sqrt{x^2} &= \pm\sqrt{\dfrac{5}{3}} \end{aligned}
x1=+53x_1 = +\sqrt{\dfrac{5}{3}}x2=−53x_2 = -\sqrt{\dfrac{5}{3}}
Mixtas
ax2+bx=0ax^2 + bx = 0 7x2−8x=6x−3x27x2+3x2−8x−6x=010x2−14x=02x(5x−7)=0\begin{aligned} 7x^2 - 8x &= 6x - 3x^2 \\ 7x^2 + 3x^2 - 8x - 6x &= 0 \\ 10x^2 - 14x &= 0 \\ 2x(5x - 7) &= 0 \end{aligned}
x1=0x_1 = 0x2=75x_2 = \dfrac{7}{5}

Otro caso de ecuación pura

2x2−3=5x2+92x2−5x2=9+3−3x2=12x2=12−3x2=−4\begin{aligned} 2x^2 - 3 &= 5x^2 + 9 \\ 2x^2 - 5x^2 &= 9 + 3 \\ -3x^2 &= 12 \\ x^2 &= \dfrac{12}{-3} \\ x^2 &= -4 \end{aligned}

El cuadrado da negativo: no existe solución en R\mathbb{R}.

Ejercicios

Bloque 1
7x2−4x=6x+2x27x^2 - 4x = 6x + 2x^2
4x2−5=3x2−24x^2 - 5 = 3x^2 - 2
Bloque 2
8x2−3x=−x+5x28x^2 - 3x = -x + 5x^2
−2x2+3=5x2−8-2x^2 + 3 = 5x^2 - 8
Bloque 3
3x−4x2=8x2−4x3x - 4x^2 = 8x^2 - 4x
−2+3x2=6+9x2-2 + 3x^2 = 6 + 9x^2

Respuestas

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Bloque 1

7x2−4x=6x+2x27x^2 - 4x = 6x + 2x^2
7x2−2x2−4x−6x=05x2−10x=05x(x−2)=0\begin{aligned} 7x^2 - 2x^2 - 4x - 6x &= 0 \\ 5x^2 - 10x &= 0 \\ 5x(x - 2) &= 0 \end{aligned}
5x=0x=0\begin{aligned} 5x &= 0 \\ x &= 0 \end{aligned} x1=0x_1 = 0
x−2=0x=2\begin{aligned} x - 2 &= 0 \\ x &= 2 \end{aligned} x2=2x_2 = 2
4x2−5=3x2−24x^2 - 5 = 3x^2 - 2
4x2−3x2=−2+5x2=3x2=±3\begin{aligned} 4x^2 - 3x^2 &= -2 + 5 \\ x^2 &= 3 \\ \sqrt{x^2} &= \pm\sqrt{3} \end{aligned}
x1=+3x_1 = +\sqrt{3}x2=−3x_2 = -\sqrt{3}

Bloque 2

8x2−3x=−x+5x28x^2 - 3x = -x + 5x^2
8x2−5x2−3x+x=03x2−2x=0x(3x−2)=0\begin{aligned} 8x^2 - 5x^2 - 3x + x &= 0 \\ 3x^2 - 2x &= 0 \\ x(3x - 2) &= 0 \end{aligned}
x=0\begin{aligned} x &= 0 \end{aligned} x1=0x_1 = 0
3x−2=03x=2x=23\begin{aligned} 3x - 2 &= 0 \\ 3x &= 2 \\ x &= \dfrac{2}{3} \end{aligned} x2=23x_2 = \dfrac{2}{3}
−2x2+3=5x2−8-2x^2 + 3 = 5x^2 - 8
−2x2−5x2=−8−3−7x2=−11x2=−11−7=117x2=±117\begin{aligned} -2x^2 - 5x^2 &= -8 - 3 \\ -7x^2 &= -11 \\ x^2 &= \dfrac{-11}{-7} = \dfrac{11}{7} \\ \sqrt{x^2} &= \pm\sqrt{\dfrac{11}{7}} \end{aligned}
x1=+117x_1 = +\sqrt{\dfrac{11}{7}}x2=−117x_2 = -\sqrt{\dfrac{11}{7}}

Bloque 3

3x−4x2=8x2−4x3x - 4x^2 = 8x^2 - 4x
−4x2−8x2+3x+4x=0−12x2+7x=0x(−12x+7)=0\begin{aligned} -4x^2 - 8x^2 + 3x + 4x &= 0 \\ -12x^2 + 7x &= 0 \\ x(-12x + 7) &= 0 \end{aligned}
x=0\begin{aligned} x &= 0 \end{aligned} x1=0x_1 = 0
−12x+7=0−12x=−7x=−7−12=712\begin{aligned} -12x + 7 &= 0 \\ -12x &= -7 \\ x &= \dfrac{-7}{-12} = \dfrac{7}{12} \end{aligned} x2=712x_2 = \dfrac{7}{12}
−2+3x2=6+9x2-2 + 3x^2 = 6 + 9x^2
3x2−9x2=6+2−6x2=8x2=−86\begin{aligned} 3x^2 - 9x^2 &= 6 + 2 \\ -6x^2 &= 8 \\ x^2 &= -\dfrac{8}{6} \end{aligned}

El cuadrado da negativo: no existe solución en R\mathbb{R}.

Ejercicios extra

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Bloque 1
6x2−9x=2x2+3x6x^2 - 9x = 2x^2 + 3x
5x2−7=2x2+55x^2 - 7 = 2x^2 + 5
4x+3x2=9x−2x24x + 3x^2 = 9x - 2x^2
7x2+2=3x2+387x^2 + 2 = 3x^2 + 38
2x2−5x=−3x+6x22x^2 - 5x = -3x + 6x^2
Bloque 2
2x2+7=9−3x22x^2 + 7 = 9 - 3x^2
10x2+4x=6x−2x210x^2 + 4x = 6x - 2x^2
−3x2+5=2x2−15-3x^2 + 5 = 2x^2 - 15
3x−5x2=7x2+8x3x - 5x^2 = 7x^2 + 8x
6x2+11=2x2−56x^2 + 11 = 2x^2 - 5
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