979 Вправа 979Алгебра 8 клас Бевз 2025 · § 19 · стор. 184

979. Solve the biquadratic equation. a) x⁴ − 10x² + 9 = 0; b) x⁴ − 26x² + 25 = 0; c) x⁴ + 3x² − 4 = 0; d) x⁴ − 4x² − 5 = 0; e) 9x⁴ − 10x² + 1 = 0; f) 5x⁴ − 11x² + 2 = 0.

Заміна ${\displaystyle x^2=}$${\displaystyle t}$, ${\displaystyle t\ge 0}$.

a) ${\displaystyle x^4-10x^2+9=}$${\displaystyle 0}$;

заміна ${\displaystyle x^2=}$${\displaystyle t}$, ${\displaystyle t\ge 0}$:

${\displaystyle t^2-10t+9=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -10,c=}$${\displaystyle 9}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-10)}^2-4\cdot 1\cdot 9=}$${\displaystyle 100-36=}$${\displaystyle 64}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{10\pm \sqrt{64}}{2}=}$${\displaystyle \frac{10\pm 8}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{10+8}{2}=}$${\displaystyle 9}$;

${\displaystyle t_2=}$${\displaystyle \frac{10-8}{2}=}$${\displaystyle 1}$;

${\displaystyle x^2=}$${\displaystyle 9}$;

${\displaystyle x=}$${\displaystyle \pm 3}$;

${\displaystyle x^2=}$${\displaystyle 1}$;

${\displaystyle x=}$${\displaystyle \pm 1}$;

b) ${\displaystyle x^4-26x^2+25=}$${\displaystyle 0}$;

заміна ${\displaystyle x^2=}$${\displaystyle t}$, ${\displaystyle t\ge 0}$:

${\displaystyle t^2-26t+25=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -26,c=}$${\displaystyle 25}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-26)}^2-4\cdot 1\cdot 25=}$${\displaystyle 676-100=}$${\displaystyle 576}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{26\pm \sqrt{576}}{2}=}$${\displaystyle \frac{26\pm 24}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{26+24}{2}=}$${\displaystyle 25}$;

${\displaystyle t_2=}$${\displaystyle \frac{26-24}{2}=}$${\displaystyle 1}$;

${\displaystyle x^2=}$${\displaystyle 25}$;

${\displaystyle x=}$${\displaystyle \pm 5}$;

${\displaystyle x^2=}$${\displaystyle 1}$;

${\displaystyle x=}$${\displaystyle \pm 1}$;

c) ${\displaystyle x^4+3x^2-4=}$${\displaystyle 0}$;

заміна ${\displaystyle x^2=}$${\displaystyle t}$, ${\displaystyle t\ge 0}$:

${\displaystyle t^2+3t-4=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 3,c=}$${\displaystyle -4}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle 3^2-4\cdot 1\cdot (-4)=}$${\displaystyle 9+16=}$${\displaystyle 25}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{-3\pm \sqrt{25}}{2}=}$${\displaystyle \frac{-3\pm 5}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{-3+5}{2}=}$${\displaystyle 1}$;

${\displaystyle t_2=}$${\displaystyle \frac{-3-5}{2}=}$${\displaystyle -4}$;

${\displaystyle t=}$${\displaystyle -4<0}$ — не підходить;

${\displaystyle x^2=}$${\displaystyle 1}$;

${\displaystyle x=}$${\displaystyle \pm 1}$;

d) ${\displaystyle x^4-4x^2-5=}$${\displaystyle 0}$;

заміна ${\displaystyle x^2=}$${\displaystyle t}$, ${\displaystyle t\ge 0}$:

${\displaystyle t^2-4t-5=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -4,c=}$${\displaystyle -5}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-4)}^2-4\cdot 1\cdot (-5)=}$${\displaystyle 16+20=}$${\displaystyle 36}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{4\pm \sqrt{36}}{2}=}$${\displaystyle \frac{4\pm 6}{2}}$;

${\displaystyle t_1=}$${\displaystyle \frac{4+6}{2}=}$${\displaystyle 5}$;

${\displaystyle t_2=}$${\displaystyle \frac{4-6}{2}=}$${\displaystyle -1}$;

${\displaystyle t=}$${\displaystyle -1<0}$ — не підходить;

${\displaystyle x^2=}$${\displaystyle 5}$;

${\displaystyle x=}$${\displaystyle \pm \sqrt{5}}$;

e) ${\displaystyle 9x^4-10x^2+1=}$${\displaystyle 0}$;

заміна ${\displaystyle x^2=}$${\displaystyle t}$, ${\displaystyle t\ge 0}$:

${\displaystyle 9t^2-10t+1=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 9,b=}$${\displaystyle -10,c=}$${\displaystyle 1}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-10)}^2-4\cdot 9\cdot 1=}$${\displaystyle 100-36=}$${\displaystyle 64}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{10\pm \sqrt{64}}{18}=}$${\displaystyle \frac{10\pm 8}{18}}$;

${\displaystyle t_1=}$${\displaystyle \frac{10+8}{18}=}$${\displaystyle 1}$;

${\displaystyle t_2=}$${\displaystyle \frac{10-8}{18}=}$${\displaystyle \frac{1}{9}}$;

${\displaystyle x^2=}$${\displaystyle 1}$;

${\displaystyle x=}$${\displaystyle \pm 1}$;

${\displaystyle x^2=}$${\displaystyle \frac{1}{9}}$;

${\displaystyle x=}$${\displaystyle \pm \frac{1}{3}}$;

f) ${\displaystyle 5x^4-11x^2+2=}$${\displaystyle 0}$;

заміна ${\displaystyle x^2=}$${\displaystyle t}$, ${\displaystyle t\ge 0}$:

${\displaystyle 5t^2-11t+2=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 5,b=}$${\displaystyle -11,c=}$${\displaystyle 2}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-11)}^2-4\cdot 5\cdot 2=}$${\displaystyle 121-40=}$${\displaystyle 81}$;

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{11\pm \sqrt{81}}{10}=}$${\displaystyle \frac{11\pm 9}{10}}$;

${\displaystyle t_1=}$${\displaystyle \frac{11+9}{10}=}$${\displaystyle 2}$;

${\displaystyle t_2=}$${\displaystyle \frac{11-9}{10}=}$${\displaystyle 0,2}$;

${\displaystyle x^2=}$${\displaystyle 2}$;

${\displaystyle x=}$${\displaystyle \pm \sqrt{2}}$;

${\displaystyle x^2=}$${\displaystyle 0,2=}$${\displaystyle \frac{1}{5}}$;

${\displaystyle x=}$${\displaystyle \pm \sqrt{\frac{1}{5}}=}$${\displaystyle \pm \frac{\sqrt{5}}{5}}$.