1. Given the expression:
[tex]\mleft(x+2\mright)\mleft(x-4\mright)[/tex]
You can use the FOIL method to multiply the binomials. Remember that the FOIL method is:
[tex](a+b)\mleft(c+d\mright)=ac+ad+bc+bd[/tex]
Then, you get:
[tex]\begin{gathered} =(x)(x)-(x)(4)+(2)(x)-(2)(4) \\ =x^2-4x+2x^{}-8 \end{gathered}[/tex]
Adding the like terms, you get:
[tex]=x^2-2x-8[/tex]
2. Given:
[tex]x^2-6x+5[/tex]
You have to complete the square:
- Identify the coefficient of the x-term". In this case, this is -6.
- Divide -6 by 2 and square the result:
[tex](\frac{-6}{2})^2=(-3)^2=9[/tex]
- Now add 9 to the polynomial and also subtract 9 from the polynomial:
[tex]=x^2-6x+(9)+5-(9)[/tex]
- Finally, simplifying and completing the square, you get:
[tex]=(x-3)^2-4[/tex]
3. Given the expression:
[tex]\mleft(x+3\mright)^2-7[/tex]
You can simplify it as follows:
- Apply:
[tex](a+b)^2=a^2+2ab+b^2[/tex]
In this case:
[tex]\begin{gathered} a=x \\ b=3 \end{gathered}[/tex]
Then:
[tex]\begin{gathered} =\lbrack(x)^2+(2)(x)(3)+(3)^2\rbrack-7 \\ =\lbrack x^2+6x+9\rbrack-7 \end{gathered}[/tex]
- Adding the like terms, you get:
[tex]=x^2+6x+2[/tex]
4. Given:
[tex]x^2-8x+15[/tex]
You need to complete the square by following the procedure used in expression 2.
In this case, the coefficient of the x-term is:
[tex]b=-8[/tex]
Then:
[tex](\frac{-8}{2})^2=(-4)^2=16[/tex]
By Completing the square, you get:
[tex]\begin{gathered} =x^2-8x+(16)+15-(16) \\ =(x-4)^2-1 \end{gathered}[/tex]
Therefore, the answer is:
