Algebra II/Quadratic Functions

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 $x=-5$± $2i$√$5$.

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