987 Вправа 987Алгебра 8 клас Бевз 2025 · § 19 · стор. 185

987. Розв’яжи рівняння. а) (2t − 3)(5t + 2) + (3t − 1)(4t + 2) = 10t² − 5; б) (3n − 2)(3n + 2) − (2n − 3)² = 3n(n + 7) − 17; в) √2b² − 3b + √2 = 0; г) c² − √6c + 1,5 = 0.

а) ${\displaystyle (2t-3)(5t+2)+(3t-1)(4t+2)=}$${\displaystyle 10t^2-5}$;

${\displaystyle 10t^2-11t-6+12t^2+2t-2=}$${\displaystyle 10t^2-5}$;

${\displaystyle 22t^2-9t-8-10t^2+5=}$${\displaystyle 0}$;

${\displaystyle 12t^2-9t-3=}$${\displaystyle 0\mid :3}$;

${\displaystyle 4t^2-3t-1=}$${\displaystyle 0}$;

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

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

${\displaystyle D>0}$, два корені:

${\displaystyle t_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{3\pm \sqrt{25}}{8}=}$${\displaystyle \frac{3\pm 5}{8}}$;

${\displaystyle t_1=}$${\displaystyle \frac{3+5}{8}=}$${\displaystyle 1}$;

${\displaystyle t_2=}$${\displaystyle \frac{3-5}{8}=}$${\displaystyle -0,25}$;

б) ${\displaystyle (3n-2)(3n+2)-{(2n-3)}^2=}$${\displaystyle 3n(n+7)-17}$;

${\displaystyle 9n^2-4-(4n^2-12n+9)=}$${\displaystyle 3n^2+21n-17}$;

${\displaystyle 5n^2+12n-13=}$${\displaystyle 3n^2+21n-17}$;

${\displaystyle 5n^2+12n-13-3n^2-21n+17=}$${\displaystyle 0}$;

${\displaystyle 2n^2-9n+4=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 2,b=}$${\displaystyle -9,c=}$${\displaystyle 4}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-9)}^2-4\cdot 2\cdot 4=}$${\displaystyle 81-32=}$${\displaystyle 49}$;

${\displaystyle D>0}$, два корені:

${\displaystyle n_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{9\pm \sqrt{49}}{4}=}$${\displaystyle \frac{9\pm 7}{4}}$;

${\displaystyle n_1=}$${\displaystyle \frac{9+7}{4}=}$${\displaystyle 4}$;

${\displaystyle n_2=}$${\displaystyle \frac{9-7}{4}=}$${\displaystyle 0,5}$;

в) ${\displaystyle \sqrt{2}{b}^2-3b+\sqrt{2}=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle \sqrt{2},b=}$${\displaystyle -3,c=}$${\displaystyle \sqrt{2}}$;

${\displaystyle D=}$${\displaystyle {(-3)}^2-4\cdot \sqrt{2}\cdot \sqrt{2}=}$${\displaystyle 9-8=}$${\displaystyle 1}$;

${\displaystyle b_{1,2}=}$${\displaystyle \frac{3\pm 1}{2\sqrt{2}}}$;

${\displaystyle b_1=}$${\displaystyle \frac{4}{2\sqrt{2}}=}$${\displaystyle \frac{2}{\sqrt{2}}=}$${\displaystyle \sqrt{2}}$;

${\displaystyle b_2=}$${\displaystyle \frac{2}{2\sqrt{2}}=}$${\displaystyle \frac{1}{\sqrt{2}}=}$${\displaystyle \frac{\sqrt{2}}{2}}$;

г) ${\displaystyle c^2-\sqrt{6}c+1,5=}$${\displaystyle 0}$;

${\displaystyle D=}$${\displaystyle {(-\sqrt{6})}^2-4\cdot 1\cdot 1,5=}$${\displaystyle 6-6=}$${\displaystyle 0}$, один корінь:

${\displaystyle c=}$${\displaystyle \frac{\sqrt{6}}{2}}$;

${\displaystyle c=}$${\displaystyle \frac{\sqrt{6}}{2}}$.