986 Вправа 986Алгебра 8 клас Бевз 2025 · § 19 · стор. 185

986. Розв’яжи рівняння. а) (2,5x − 7)(2x + 3) + 3x + 4 = (4x − 9)(1,5x + 1); б) (3z − 5)(4z + 1) + (2z + 3)(5z − 4) = 6z(3 + 2z) − 11; в) x² + 2√2x + 1 = 0; г) x² − 3√2x + 4 = 0.

а) ${\displaystyle (2,5x-7)(2x+3)+3x+4=}$${\displaystyle (4x-9)(1,5x+1)}$;

${\displaystyle 5x^2+7,5x-14x-21+3x+4=}$${\displaystyle 6x^2+4x-13,5x-9}$;

${\displaystyle 5x^2-3,5x-17=}$${\displaystyle 6x^2-9,5x-9}$;

${\displaystyle 0=}$${\displaystyle 6x^2-5x^2-9,5x+3,5x-9+17}$;

${\displaystyle 0=}$${\displaystyle x^2-6x+8}$;

${\displaystyle x^2-6x+8=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -6,c=}$${\displaystyle 8}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-6)}^2-4\cdot 1\cdot 8=}$${\displaystyle 36-32=}$${\displaystyle 4}$;

${\displaystyle D>0}$, два корені:

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

${\displaystyle x_1=}$${\displaystyle \frac{6+2}{2}=}$${\displaystyle 4}$;

${\displaystyle x_2=}$${\displaystyle \frac{6-2}{2}=}$${\displaystyle 2}$;

б) ${\displaystyle (3z-5)(4z+1)+(2z+3)(5z-4)=}$${\displaystyle 6z(3+2z)-11}$;

${\displaystyle 12z^2-17z-5+10z^2+7z-12=}$${\displaystyle 18z+12z^2-11}$;

${\displaystyle 22z^2-10z-17=}$${\displaystyle 12z^2+18z-11}$;

${\displaystyle 10z^2-28z-6=}$${\displaystyle 0\mid :2}$;

${\displaystyle 5z^2-14z-3=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 5,b=}$${\displaystyle -14,c=}$${\displaystyle -3}$;

${\displaystyle D=}$${\displaystyle b^2-4ac=}$${\displaystyle {(-14)}^2-4\cdot 5\cdot (-3)=}$${\displaystyle 196+60=}$${\displaystyle 256}$;

${\displaystyle D>0}$, два корені:

${\displaystyle z_{1,2}=}$${\displaystyle \frac{-b\pm \sqrt{D}}{2a}=}$${\displaystyle \frac{14\pm \sqrt{256}}{10}=}$${\displaystyle \frac{14\pm 16}{10}}$;

${\displaystyle z_1=}$${\displaystyle \frac{14+16}{10}=}$${\displaystyle 3}$;

${\displaystyle z_2=}$${\displaystyle \frac{14-16}{10}=}$${\displaystyle -0,2}$;

в) ${\displaystyle x^2+2\sqrt{2}x+1=}$${\displaystyle 0}$;

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle 2\sqrt{2},c=}$${\displaystyle 1}$;

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

${\displaystyle x_{1,2}=}$${\displaystyle \frac{-2\sqrt{2}\pm 2}{2}=}$${\displaystyle -\sqrt{2}\pm 1}$;

${\displaystyle x_1=}$${\displaystyle -\sqrt{2}+1}$;

${\displaystyle x_2=}$${\displaystyle -\sqrt{2}-1}$;

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

${\displaystyle a=}$${\displaystyle 1,b=}$${\displaystyle -3\sqrt{2},c=}$${\displaystyle 4}$;

${\displaystyle D=}$${\displaystyle {(-3\sqrt{2})}^2-4\cdot 1\cdot 4=}$${\displaystyle 18-16=}$${\displaystyle 2}$;

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

${\displaystyle x_1=}$${\displaystyle \frac{4\sqrt{2}}{2}=}$${\displaystyle 2\sqrt{2}}$;

${\displaystyle x_2=}$${\displaystyle \frac{2\sqrt{2}}{2}=}$${\displaystyle \sqrt{2}}$.