A quadratic function is represented by the following equation:

$y=ax$ 2$+bx+c$ • $ax$ 2 = Quadratic Term
• $bx$ = Linear Term
• $c$ = Constant Term

## Solving Quadratic Functions by Factoring

1. $x$ 2 $+7x+6$ • Factor them: We get $(x+6)(x+1)$ .
• The linear terms must add to make 7.
• The constant terms needs to multiply to make 6.
• Set them out as problems to solve:
• $x+6=0$ → x = -6
• $x+1=0$ → x = -1
• Your answers are $-6$ and $-1$ .

2. $x$ 2 $-9$ .

• Factor them: We get $(x-3)(x+3)$ .
• Set them out as problems to solve:
• $x+3=0$ → x = -3
• $x-3=0$ → x = 3
• Your answers are $3$ and $-3$ .

3. $2x$ 2$+12x+10$ • Divide all of the terms by the GCF: 2: We get a new problem to deal with, which is $x$ 2$+6x+5$ .
• Factor them: We get $2(x+5)(x+1)$ .
• Set them out as problems to solve:
• $x+5=0$ → x = -5
• $x+1=0$ → x = -1
• Your answers are $-5$ and $-1$ .

## Solving Quadratic Factors by Completing the Square

1. $x$ 2 $+8x$ __ = $19$ • Take the Linear Term and divide it by $2$ : We get $4$ .
• We take this number, $4$ , and square it: We get $16$ .
• We add $16$ to $19$ : We get $35$ .
• We now have: $(x+4)$ 2 $=35$ .
• We square both sides: We get $x+4=$ $35$ .
• We minus 4 to the other side. Here is our answer.: $x=-4$ ± √$35$ .

2. $x$ 2 $+x$ + ___

• Take the Linear Term and divide it by $2$ : We get ${\tfrac {1}{2}}$ .
• We take this number, ${\tfrac {1}{2}}$ , and square it: We get ${\tfrac {1}{4}}$ .
• We have our answer: $x$ 2 $+x$ + ${\tfrac {1}{4}}$ .

3. $x$ 2 $+45$ = $10x$ • Rearrange this problem so that it matches the standard format for a quadratic equation: We switch the $45$ and the $10x$ around, forming our new problem: $x$ 2 $-10x$ = $-45$ .
• Divide the Linear Tearm, $bx$ , by $2$ : This gives us $5$ .
• Square the $5$ : This gives us $25$ .
• Add the $25$ to $-45$ : This brings our problem to ($x$ $-5)$ 2 = $-20$ .
• Square both sides of the problem: This brings us to $x-5=$ i√$20$ .
• Find the square root of $20$ (don't forget the $i$ ) and then add $5$ to the opposite side to find your answer: Our final answer is $x=-5$ ± $2i$ $5$ .