Answer :
[tex]x^2-2x-24=0\\\\a=1;\ b=-2;\ c=-24\\\\\Delta=b^2-4ac\\\\\Delta=(-2)^2-4\cdot1\cdot(-24)=4+96=100\\\\x_1=\frac{-b-\sqrt\Delta}{2a};\ x_2=\frac{-b+\sqrt\Delta}{2a}\\\\x_1=\frac{2-\sqrt{100}}{2\cdot1}=\frac{2-10}{2}=\frac{-8}{2}=-4\\\\x_2=\frac{2+\sqrt{100}}{2\cdot1}=\frac{2+10}{2}=\frac{12}{2}=6[/tex]
There are another way to solve a quadratic equation we call, Sum and product.
Let's see how we can do this...
[tex]x^2-2x-24=0[/tex]
[tex]Sum=-\frac{b}{a}[/tex]
[tex]Product=\frac{c}{a}[/tex]
therefore
[tex]Sum=-\frac{(-2)}{1}=2[/tex]
[tex]Product=\frac{-24}{1}=-24[/tex]
now we have to pick up 2 numbers that the sum should be 2 and the product should be -24, we just have to think a little.
Let's try -3 and 8, for example.
[tex]8+(-3)=5[/tex]
[tex]8*(-3)=-24[/tex]
Doesn't work.
Let's try now 4 and -6.
[tex]4+(-6)=-2[/tex]
[tex]4*(-6)=-24[/tex]
Can you see here, that we have to change the signal?!
therefore
Let's try -4 and 6
[tex]6+(-4)=2[/tex]
[tex]6*(-4)=-24[/tex]
[tex]Sum=2[/tex]
and
[tex]Product=-24[/tex]
Them it works.
[tex]\boxed{\boxed{x_1=-4~~and~~x_2=6}}[/tex]
Let's see how we can do this...
[tex]x^2-2x-24=0[/tex]
[tex]Sum=-\frac{b}{a}[/tex]
[tex]Product=\frac{c}{a}[/tex]
therefore
[tex]Sum=-\frac{(-2)}{1}=2[/tex]
[tex]Product=\frac{-24}{1}=-24[/tex]
now we have to pick up 2 numbers that the sum should be 2 and the product should be -24, we just have to think a little.
Let's try -3 and 8, for example.
[tex]8+(-3)=5[/tex]
[tex]8*(-3)=-24[/tex]
Doesn't work.
Let's try now 4 and -6.
[tex]4+(-6)=-2[/tex]
[tex]4*(-6)=-24[/tex]
Can you see here, that we have to change the signal?!
therefore
Let's try -4 and 6
[tex]6+(-4)=2[/tex]
[tex]6*(-4)=-24[/tex]
[tex]Sum=2[/tex]
and
[tex]Product=-24[/tex]
Them it works.
[tex]\boxed{\boxed{x_1=-4~~and~~x_2=6}}[/tex]
