Eq2 equations

For each of the following problems, determine the roots of the equation. Given the roots, sketch the graph and explain how your sketch matches the roots given and the form of the equation:

g(x)=36x²-12x+5
h(x)=x²-4x+20
f(x)=4x²-24x+45
p(x)=9x²-36x+40
g(x)=x²-2x+10
g(x)=x2-x-6

Correct answer:

g₁ = 0.166667+0.333333i
g₂ = 0.166667-0.333333i
h₁ = 2+4i
h₂ = 2-4i

Step-by-step explanation:

$$\begin{gathered} g(x)=36x^2-12x+5 \\ 36g^2-12g+5=0 \\ 36g^2-12g+5=0 \\ a=36;b=-12;c=5 \\ D=b^2-4ac=12^2-4\cdot 36\cdot 5=-576 \\ D<0 \\ {g}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{12\pm \sqrt{-576}}{72}=\frac{12\pm 24\sqrt{-1}}{72} \\ {g}_{1,2}=\frac{12\pm 24\text{i}}{72} \\ {g}_{1,2}=0.166667\pm 0.333333\text{i} \\ {g}_1=0.166667+0.33333333333333\text{i} \\ {g}_2=0.166667-0.33333333333333\text{i} \\ {g}_1=0.166667+0.333333i \end{gathered}$$

Our quadratic equation calculator calculates it.

$g_2=0.166667-0.333333i$

$$\begin{gathered} h(x)=x^2-4x+20 \\ {h}^2-4\cdot h+20=0 \\ {h}^2-4\cdot h+20=0 \\ {h}^2-4h+20=0 \\ a=1;b=-4;c=20 \\ D=b^2-4ac=4^2-4\cdot 1\cdot 20=-64 \\ D<0 \\ {h}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{4\pm \sqrt{-64}}{2}=\frac{4\pm 8\sqrt{-1}}{2} \\ {h}_{1,2}=\frac{4\pm 8\text{i}}{2} \\ {h}_{1,2}=2\pm 4\text{i} \\ {h}_1=2+4\text{i} \\ {h}_2=2-4\text{i} \\ {h}_1=2+4i \end{gathered}$$

Our quadratic equation calculator calculates it.

$h_2=2-4i$

Try another example