Quadratic Formula Practice AK

1. $a=3$, $b=-6$, $c=2$

$\eqalign{=&\dfrac{6 \pm \sqrt{(-6)^2-4(3)(2)}}{2(3)}\\=&\dfrac{6 \pm \sqrt{36-24}}{6}\\=&\dfrac{6 \pm \sqrt{12}}{6}\\=&1 \pm \dfrac{2\sqrt{3}}{6}\\=&1\pm\dfrac{\sqrt{3}}{3}}$

2. $a=1$, $b=7$, $c=-1$

$\eqalign{=&\dfrac{-7 \pm \sqrt{(7)^2-4(1)(-1)}}{2(1)}\\=&\dfrac{-7 \pm \sqrt{49+4}}{2}\\=&\dfrac{-7 \pm \sqrt{53}}{2}}$

3. $x^2-10x+18$

$a=1$, $b=-10$, $c=18$

$\eqalign{=&\dfrac{10 \pm \sqrt{(-10)^2-4(1)(18)}}{2(1)}\\=&\dfrac{10 \pm \sqrt{100-72}}{2}\\=&\dfrac{10 \pm \sqrt{28}}{2}\\=&5\pm\sqrt{7}}$

4. $4x^2-4x-39$

$a=4$, $b=-4$, $c=-39$

$\eqalign{=&\dfrac{4 \pm \sqrt{(-4)^2-4(4)(-39)}}{2(4)}\\=&\dfrac{4 \pm \sqrt{16+624}}{8}\\=&\dfrac{4 \pm \sqrt{640}}{8}\\=&\dfrac{4\pm8\sqrt{10}}{8}\\=&\dfrac{1}{2}\pm \sqrt{10}}$

5. $x^2+3x-6$

$a=1$, $b=3$, $c=-6$

$\eqalign{=&\dfrac{-3 \pm \sqrt{(3)^2-4(1)(-6)}}{2(1)}\\=&\dfrac{-3 \pm \sqrt{9+24}}{2}\\=&\dfrac{-3 \pm \sqrt{33}}{2}}$

6. $a^2+4a-20$

$a=1$, $b=4$, $c=-20$

$\eqalign{=&\dfrac{-4 \pm \sqrt{(4)^2-4(1)(-20)}}{2(1)}\\=&\dfrac{-4 \pm \sqrt{16+80}}{2}\\=&\dfrac{-4 \pm \sqrt{96}}{2}\\=&\dfrac{-4\pm 4\sqrt{6}}{2}\\=&-2\pm2\sqrt{6}}$

7. $a=.5$, $b=.2$, $c=-.1$

$\eqalign{=&\dfrac{-.2 \pm \sqrt{(.2)^2-4(.5)(-.1)}}{2(.5)}\\=&\dfrac{-.2 \pm \sqrt{.04+.2}}{1}\\=&\dfrac{-.2 \pm \sqrt{.24}}{1}\\=&\dfrac{-.2 \pm .2\sqrt{.6}}{1}\\=&-.2 \pm .2 \sqrt{.6}}$

8. $a=1$, $b=-\dfrac{5}{6}$, $c=3$

$\eqalign{=&\dfrac{\dfrac{5}{6} \pm \sqrt{(-\dfrac{5}{6})^2-4(1)(3)}}{2(1)}\\=&\dfrac{\dfrac{5}{6} \pm \sqrt{\dfrac{25}{36}-12}}{2}\\=&\dfrac{\dfrac{5}{6} \pm \sqrt{-11\dfrac{11}{36}}}{2}}$

9. $a=\dfrac{1}{3}$, $b=5$, $c=-2$

$\eqalign{=&\dfrac{-5 \pm \sqrt{(5)^2-4(\dfrac{1}{3})(-2)}}{2(\dfrac{1}{3})}\\=&\dfrac{-5 \pm \sqrt{25+\dfrac{8}{3}}}{\dfrac{2}{3}}\\=&\dfrac{-5 \pm \sqrt{27 \dfrac{2}{3}}}{\dfrac{2}{3}}\\=&\dfrac{-15 \pm 3 \sqrt{27\dfrac{2}{3}}}{2}}$

10. $a=1.2$, $b=20$, $c=-3$

$\eqalign{=&\dfrac{-20 \pm \sqrt{(20)^2-4(1.2)(-3)}}{2(1.2)}\\=&\dfrac{-20 \pm \sqrt{400+14.4}}{2.4}\\=&\dfrac{-20 \pm \sqrt{414.4}}{2.4}\\=&\dfrac{-20 \pm \sqrt{414.4}}{\dfrac{24}{10}}\\=&\dfrac{-200 \pm 10\sqrt{414.4}}{24}\\&=\dfrac{-100 \pm 5 \sqrt{414.4}}{12}}$