Question

Записать формулу корней квадратного уравнения х²+2mх+с=0, решить с помощью этой формулы уравнение:

1. х²+10х+24=0

2. х²-23х-51=0

Answer 1

$x^2+2mx+c=0$

$x_1=frac{-2m+sqrt{(2m)^2-4c}}{2}=frac{-2m+sqrt{4m^2-4c}}{2}=frac{-2m+sqrt{4(m^2-c)}}{2}=frac{-2m+2sqrt{m^2-c}}{2}=-m+sqrt{m^2-c}=<var>sqrt{m^2-c}-m</var>$

 

$x_2=frac{-2m-sqrt{(2m)^2-4c}}{2}=frac{-2m-sqrt{4m^2-4c}}{2}=frac{-2m-sqrt{4(m^2-c)}}{2}=frac{-2m-2sqrt{m^2-c}}{2}=-m-sqrt{m^2-c}=-(m </var><var>sqrt{m^2-c}</var>)$</p> <p> </p> <p>1) <img src=[/tex]x^2+10x+24=0" title="x_2=frac{-2m-sqrt{(2m)^2-4c}}{2}=frac{-2m-sqrt{4m^2-4c}}{2}=frac{-2m-sqrt{4(m^2-c)}}{2}=frac{-2m-2sqrt{m^2-c}}{2}=-m-sqrt{m^2-c}=-(m+sqrt{m^2-c})" title="x^2+10x+24=0" title="x_2=frac{-2m-sqrt{(2m)^2-4c}}{2}=frac{-2m-sqrt{4m^2-4c}}{2}=frac{-2m-sqrt{4(m^2-c)}}{2}=frac{-2m-2sqrt{m^2-c}}{2}=-m-sqrt{m^2-c}=-(m+sqrt{m^2-c})" alt="x^2+10x+24=0" title="x_2=frac{-2m-sqrt{(2m)^2-4c}}{2}=frac{-2m-sqrt{4m^2-4c}}{2}=frac{-2m-sqrt{4(m^2-c)}}{2}=frac{-2m-2sqrt{m^2-c}}{2}=-m-sqrt{m^2-c}=-(m+sqrt{m^2-c})" />

 

1) $x_2=frac{-2m-sqrt{(2m)^2-4c}}{2}=frac{-2m-sqrt{4m^2-4c}}{2}=frac{-2m-sqrt{4(m^2-c)}}{2}=frac{-2m-2sqrt{m^2-c}}{2}=-m-sqrt{m^2-c}=-(m+</var><var>sqrt{m^2-c}</var>)$

 

1) $<var>x^2 10x 24=0$

$x^2+2cdot5x+24=0$

$x_1=<var>sqrt{m^2-c}-m</var>$

$x_1=<var>sqrt{5^2-24}-5</var></var>=1-5=-4$

$x^2+2cdot5x+24=0$

$x_1=<var>sqrt{m^2-c}-m</var>$

$<var>x^2+10x+24=0$

$x^2+2cdot5x+24=0$

$x_1=<var>sqrt{m^2-c}-m</var>$

$x_1=<var>sqrt{5^2-24}-5</var></var>=1-5=-4$

$<var>x_2=<var><var>-(m+</var><var>sqrt{m^2-c}</var>)</var>$

$x_1=<var>sqrt{5^2-24}-5</var></var>=1-5=-4$

$<var>x_2=<var><var>-(m </var><var>sqrt{m^2-c}</var>)</var>$

$x_2=<var><var>-(5 </var><var>sqrt{5^2-24}</var>)</var></var>=-(5 1)=-6$

$<var>x_2=<var><var>-(m+</var><var>sqrt{m^2-c}</var>)</var>$

$x_2=<var><var>-(5 </var><var>sqrt{5^2-24}</var>)</var></var>=-(5 1)=-6$

Ответ: -6; -4.

 

2) $<var>x^2-23x-51=0$

Ответ: -6; -4.

 

2) $x_2=<var><var>-(5+</var><var>sqrt{5^2-24}</var>)</var></var>=-(5+1)=-6$

Ответ: -6; -4.

 

2) $<var>x^2-23x-51=0$

$x^2+2cdot(-11,5)x-51=0$

$x_1=<var>sqrt{m^2-c}-m</var>$

$x_1=<var>sqrt{(-11,5)^2-(-51)}-(-11,5)</var></var>=11,5 sqrt{frac{733}{4}}=11,5 0,5sqrt{733}$

$x^2+2cdot(-11,5)x-51=0$

$x_1=<var>sqrt{m^2-c}-m</var>$

$<var>x^2-23x-51=0$

$x^2+2cdot(-11,5)x-51=0$

$x_1=<var>sqrt{m^2-c}-m</var>$

$x_1=<var>sqrt{(-11,5)^2-(-51)}-(-11,5)</var></var>=11,5 sqrt{frac{733}{4}}=11,5 0,5sqrt{733}$

$<var>x_2=<var><var>-(m+</var><var>sqrt{m^2-c}</var>)</var>$

$x_1=<var>sqrt{(-11,5)^2-(-51)}-(-11,5)</var></var>=11,5+sqrt{frac{733}{4}}=11,5+0,5sqrt{733}$

$<var>x_2=<var><var>-(m </var><var>sqrt{m^2-c}</var>)</var>$

$<var>x_2=<var><var>-(-11,5+</var><var>sqrt{(-11,5)^2-(-51)}</var>)</var></var>=11,5-sqrt{frac{733}{4}}=11,5-0,5sqrt{733}" title="<var>x_2=<var><var>-(m+</var><var>sqrt{m^2-c}</var>)</var>" /></var></p> <p>[tex]<var>x_2=<var><var>-(-11,5+</var><var>sqrt{(-11,5)^2-(-51)}</var>)</var></var>=11,5-sqrt{frac{733}{4}}=11,5-0,5sqrt{733}" alt="<var>x_2=<var><var>-(m+</var><var>sqrt{m^2-c}</var>)</var>" /></var></p> <p>[tex]<var>x_2=<var><var>-(-11,5+</var><var>sqrt{(-11,5)^2-(-51)}</var>)</var></var>=11,5-sqrt{frac{733}{4}}=11,5-0,5sqrt{733}" /></p> <p>Ответ: [tex]<var>x_1=</var>11,5+0,5sqrt{733}$; $<var>x_2=</var>11,5-0,5sqrt{733}$.