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Matemáticas 3 · Tema 9

Completar el trinomio cuadrado perfecto

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Ejemplo

2x2−12x+10=02x^2 - 12x + 10 = 0
2x2−12x+102=02x2−6x+5=0x2−6x=−5\begin{aligned} \dfrac{2x^2 - 12x + 10}{2} &= \dfrac{0}{2} \\ x^2 - 6x + 5 &= 0 \\ x^2 - 6x &= -5 \end{aligned}
b−6-6
mitad−3-3
cuadrado de la mitad99
x2−6x+9=−5+9(x−3)(x−3)=4(x−3)2=4(x−3)2=4x−3=±2\begin{aligned} x^2 - 6x + 9 &= -5 + 9 \\ (x - 3)(x - 3) &= 4 \\ (x - 3)^2 &= 4 \\ \sqrt{(x-3)^2} &= \sqrt{4} \\ x - 3 &= \pm 2 \end{aligned}
x−3=+2x=2+3\begin{aligned} x - 3 &= +2 \\ x &= 2 + 3 \end{aligned} x1=5x_1 = 5
x−3=−2x=−2+3\begin{aligned} x - 3 &= -2 \\ x &= -2 + 3 \end{aligned} x2=1x_2 = 1

Ejercicios

Ejercicio 1
3x2−30x+27=03x^2 - 30x + 27 = 0
Ejercicio 2
3x2+6x−45=03x^2 + 6x - 45 = 0
Ejercicio 3
2x2−4x−16=02x^2 - 4x - 16 = 0
Ejercicio 4
2x2−12x−32=02x^2 - 12x - 32 = 0

Respuestas

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Ejercicio 1
3x2−30x+27=03x^2 - 30x + 27 = 0
3x2−30x+273=03x2−10x+9=0x2−10x=−9\begin{aligned} \dfrac{3x^2 - 30x + 27}{3} &= \dfrac{0}{3} \\ x^2 - 10x + 9 &= 0 \\ x^2 - 10x &= -9 \end{aligned}
b−10-10
mitad−5-5
cuadrado de la mitad2525
x2−10x+25=−9+25(x−5)(x−5)=16(x−5)2=16(x−5)2=16x−5=±4\begin{aligned} x^2 - 10x + 25 &= -9 + 25 \\ (x - 5)(x - 5) &= 16 \\ (x - 5)^2 &= 16 \\ \sqrt{(x-5)^2} &= \sqrt{16} \\ x - 5 &= \pm 4 \end{aligned}
x−5=+4x=4+5\begin{aligned} x - 5 &= +4 \\ x &= 4 + 5 \end{aligned} x1=9x_1 = 9
x−5=−4x=−4+5\begin{aligned} x - 5 &= -4 \\ x &= -4 + 5 \end{aligned} x2=1x_2 = 1
Ejercicio 2
3x2+6x−45=03x^2 + 6x - 45 = 0
3x2+6x−453=03x2+2x−15=0x2+2x=+15\begin{aligned} \dfrac{3x^2 + 6x - 45}{3} &= \dfrac{0}{3} \\ x^2 + 2x - 15 &= 0 \\ x^2 + 2x &= +15 \end{aligned}
b22
mitad11
cuadrado de la mitad11
x2+2x+1=15+1(x+1)(x+1)=16(x+1)2=16(x+1)2=16x+1=±4\begin{aligned} x^2 + 2x + 1 &= 15 + 1 \\ (x + 1)(x + 1) &= 16 \\ (x + 1)^2 &= 16 \\ \sqrt{(x+1)^2} &= \sqrt{16} \\ x + 1 &= \pm 4 \end{aligned}
x+1=+4x=4−1\begin{aligned} x + 1 &= +4 \\ x &= 4 - 1 \end{aligned} x1=3x_1 = 3
x+1=−4x=−4−1\begin{aligned} x + 1 &= -4 \\ x &= -4 - 1 \end{aligned} x2=−5x_2 = -5
Ejercicio 3
2x2−4x−16=02x^2 - 4x - 16 = 0
2x2−4x−162=02x2−2x−8=0x2−2x=+8\begin{aligned} \dfrac{2x^2 - 4x - 16}{2} &= \dfrac{0}{2} \\ x^2 - 2x - 8 &= 0 \\ x^2 - 2x &= +8 \end{aligned}
b−2-2
mitad−1-1
cuadrado de la mitad11
x2−2x+1=8+1(x−1)(x−1)=9(x−1)2=9(x−1)2=9x−1=±3\begin{aligned} x^2 - 2x + 1 &= 8 + 1 \\ (x - 1)(x - 1) &= 9 \\ (x - 1)^2 &= 9 \\ \sqrt{(x-1)^2} &= \sqrt{9} \\ x - 1 &= \pm 3 \end{aligned}
x−1=+3x=3+1\begin{aligned} x - 1 &= +3 \\ x &= 3 + 1 \end{aligned} x1=4x_1 = 4
x−1=−3x=−3+1\begin{aligned} x - 1 &= -3 \\ x &= -3 + 1 \end{aligned} x2=−2x_2 = -2
Ejercicio 4
2x2−12x−32=02x^2 - 12x - 32 = 0
2x2−12x−322=02x2−6x−16=0x2−6x=+16\begin{aligned} \dfrac{2x^2 - 12x - 32}{2} &= \dfrac{0}{2} \\ x^2 - 6x - 16 &= 0 \\ x^2 - 6x &= +16 \end{aligned}
b−6-6
mitad−3-3
cuadrado de la mitad99
x2−6x+9=16+9(x−3)(x−3)=25(x−3)2=25(x−3)2=25x−3=±5\begin{aligned} x^2 - 6x + 9 &= 16 + 9 \\ (x - 3)(x - 3) &= 25 \\ (x - 3)^2 &= 25 \\ \sqrt{(x-3)^2} &= \sqrt{25} \\ x - 3 &= \pm 5 \end{aligned}
x−3=+5x=5+3\begin{aligned} x - 3 &= +5 \\ x &= 5 + 3 \end{aligned} x1=8x_1 = 8
x−3=−5x=−5+3\begin{aligned} x - 3 &= -5 \\ x &= -5 + 3 \end{aligned} x2=−2x_2 = -2

Ejercicios extra

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Bloque 1
2x2−8x−10=02x^2 - 8x - 10 = 0
3x2+12x−15=03x^2 + 12x - 15 = 0
x2−8x+7=0x^2 - 8x + 7 = 0
2x2+4x−48=02x^2 + 4x - 48 = 0
5x2−20x−60=05x^2 - 20x - 60 = 0
Bloque 2
x2+10x+16=0x^2 + 10x + 16 = 0
4x2−8x−60=04x^2 - 8x - 60 = 0
3x2−18x+24=03x^2 - 18x + 24 = 0
x2+6x−27=0x^2 + 6x - 27 = 0
2x2+20x+42=02x^2 + 20x + 42 = 0
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