Question

Resolver por fórmula general: 25 (x+2)² = (x-7)² -81

Answer 1

Respuesta:

25(x+2)²=(x-7)²-81

25(x²+4x+4)=x²-14x+49-81

25x²+100x+100=x²+14x-32

24x²+86x+132=0

simplificamos

12x²+43x+66=0

$\frac{ - 43 + \sqrt{ {43}^{2} - 4 \times 12 \times 66 } }{2 \times 12}$

$\frac{ - 43 + \sqrt{ - 1319} }{24}$

$\frac{ - 43 + 1319i}{24}$

la otra raiz es :

$\frac{ - 43 - 1319i}{24}$

Answer 2

Tenemos la ecuación:

$25(x+2)^{2} =(x-7)^{2}-81$

La desarrollamos:

$25(x^{2} +4x+4)=x^{2} -14x+49-81\\\\25x^{2} +100x+100=x^{2} -14x-32\\\\25x^{2} +100x+100-x^{2} +14x+32=0\\\\25x^{2} -x^{2} +100x+14x+100+32=0\\\\24x^{2} +114x+132=0\\\\\frac{24x^{2} +114x+132=0}{6} \\\\4x^{2} +19x+22=0$

Ya que tenemos la ecuación desarrollada, sabemos que:

a=4, b=19, c=22

Usamos la chicharronera (fórmula general para los desentendidos):

$\frac{-b+\sqrt{b^{2}-4ac } }{2a} \\\\\frac{-b-\sqrt{b^{2}-4ac } }{2a}$

Reemplazamos datos:

$\frac{-(19)+\sqrt{(19)^{2}-4(4)(22) } }{2(4)} = \\\\\frac{-19+\sqrt{361-352 } }{8} =\\\\\frac{-19+\sqrt{9} }{8} =\\\\\frac{-19+3 }{8} =\\\\\frac{-16}{8} =\\\\-2$

x₁ vale -2

$\frac{-(19)-\sqrt{(19)^{2}-4(4)(22) } }{2(4)} = \\\\\frac{-19-\sqrt{361-352 } }{8} =\\\\\frac{-19-\sqrt{9} }{8} =\\\\\frac{-19-3 }{8} =\\\\\frac{-22}{8} =\\\\-\frac{11}{4}$

x₂ vale -(11/4)