Question

A. What is the sum of the squares of the roots of $x^2 - 5x - 4 = 0$?

B. One root of $x^2 + 12x + k = 0$ is twice the other root. Find $k.$
C. What is the sum of the roots of the quadratic $4x^2 - 4x - 4$?
D. Jimmy is trying to factor the quadratic equation $ax^2 + bx + c = 0.$ He assumes that it will factor in the form
\[ax^2 + bx + c = (Ax + B)(Cx + D),\]where $A,$ $B,$ $C,$ and $D$ are integers. If $a = 4,$ and Jimmy wants to find the value of $A,$ what are the possible values he should check, in order to find $A$?
E. Brandy is trying to factor the quadratic $3x^2 - x - 10.$ She starts by assuming that the quadratic factors as
\[3x^2 - x - 10 = (x + B)(3x + D),\]for some integers $B$ and $D.$ After some work, Brandy successfully factors the quadratic. Find the ordered pair $(B,D).$

Answer 1

A. 33

B. k=32

C. 1

D.
`\pm 1,\ \pm 2,\ \pm 4`

E.
`B=-2,\ D=5`

Explanation:

In all parts for the quadratic equation
`ax^2+bx+c=0` use Vieta's formulas

`x_1+x_2=-(b)/(a),\\ \\x_1\cdot x_2=(c)/(a),`

where
`x_1,\ x_2` are the roots of the quadratic equation.

A. For the equation
`x^2-5x-4=0,`

`x_1+x_2=5,\\ \\x_1\cdot x_2=-4.`

Then

`(x_1+x_2)^2=x_1^2+2x_1\cdot x_2+x_2^2,\\ \\5^2=x_1^2+x_2^2+2\cdot (-4),\\ \\x_1^2+x_2^2=25+8=33.`

B. One of the roots of
`x^2+12x+k=0` is twice the other root, then
`x_2=2x_1.` By the Vieta's formulas,

`x_1+x_2=3x_1=-12,\\ \\x_1\cdot x_2=2x_1^2=k.`

Then
`x_1=-4` and
`k=2x_1^2=2\cdot (-4)^2=2\cdot 16=32.`

C. The sum of the roots of the quadratic
`4x^2-4x-4` is
`-(b)/(c)=-(-4)/(4)=1.`

D. Note that

`(Ax+B)(Cx+D)=ACx^2+x(AD+BC)+BD,`

then
`AC=a=4.` If
`A,\ B,\ C,\ D` are integers, then you should check
`A=\pm 1,\ \pm 2,\ \pm 4.`

E. Consider
`3x^2 - x - 10 = (x + B)(3x + D).` Note that

`x_1+x_2=(1)/(3),\\ \\x_1\cdot x_2=-(10)/(3).`

Then

`x_1=2,\ x_2=-(5)/(3).`

Then
`3x^2 - x - 10 = (x -2)(3x+5),` hence
`B=-2,\ D=5.`