Question

Solve the following quadratic equations.
(1) $\frac{1}{x+5}=\frac{1}{x^2}$
(2) $x^2-\frac{3x}{10}-\frac{1}{10}=0$
(3) ${\left(2x+3\right)}^2=25$
(4) $m^2+5m+5=0$
(5) $5m^2+2m+1=0$
(6) $x^2-4x-3=0$

Answer 1

(1) $\frac{1}{x+5}=\frac{1}{x^2}$
$$\begin{gathered} \Rightarrow x^2=x+5 \\ \Rightarrow x^2-x-5=0 \\ \Rightarrow x=\frac{-\left(-1\right)\pm \sqrt{{\left(-1\right)}^2-4\times 1\times \left(-5\right)}}{2\times 1} \\ \Rightarrow x=\frac{1\pm \sqrt{1+20}}{2}=\frac{1\pm \sqrt{21}}{2} \end{gathered}$$
(2) $x^2-\frac{3x}{10}-\frac{1}{10}=0$
$$\begin{gathered} \Rightarrow 10x^2-3x-1=0 \\ \Rightarrow x=\frac{-\left(-3\right)\pm \sqrt{{\left(-3\right)}^2-4\times 10\times \left(-1\right)}}{2\times 10} \\ \Rightarrow x=\frac{3\pm \sqrt{9+40}}{20}=\frac{3\pm \sqrt{49}}{20} \\ \Rightarrow x=\frac{3\pm 7}{20} \\ \Rightarrow x=\frac{3+7}{20},\frac{3-7}{20} \\ \Rightarrow x=\frac{10}{20},\frac{-4}{20} \\ \Rightarrow x=\frac{1}{2},\frac{-1}{5} \end{gathered}$$

(3) ${\left(2x+3\right)}^2=25$
$$\begin{gathered} \Rightarrow 4x^2+9+12x=25 \\ \Rightarrow 4x^2+12x-16=0 \\ \Rightarrow x^2+3x-4=0 \\ \Rightarrow x^2+4x-x-4=0 \\ \Rightarrow x\left(x+4\right)-1\left(x+4\right)=0 \\ \Rightarrow \left(x-1\right)\left(x+4\right)=0 \\ \Rightarrow x=1,-4 \end{gathered}$$

(4) $m^2+5m+5=0$
$$\begin{gathered} \Rightarrow m=\frac{-5\pm \sqrt{{\left(5\right)}^2-4\times 1\times 5}}{2} \\ \Rightarrow m=\frac{-5\pm \sqrt{25-20}}{2} \\ \Rightarrow m=\frac{-5\pm \sqrt{5}}{2} \end{gathered}$$

(5) $5m^2+2m+1=0$
$$\begin{gathered} \Rightarrow m=\frac{-2\pm \sqrt{{\left(2\right)}^2-4\times 1\times 5}}{2} \\ \Rightarrow m=\frac{-2\pm \sqrt{4-20}}{2} \\ \Rightarrow m=\frac{-2\pm \sqrt{-16}}{2} \end{gathered}$$
So, the roots are not real as the discriminant is negative.

(6) $x^2-4x-3=0$
$$\begin{gathered} \Rightarrow x=\frac{-\left(-4\right)\pm \sqrt{{\left(-4\right)}^2-4\times 1\times \left(-3\right)}}{2\times 1} \\ \Rightarrow x=\frac{4\pm \sqrt{16+12}}{2} \\ \Rightarrow x=\frac{4\pm \sqrt{28}}{2} \\ \Rightarrow x=\frac{4\pm 2\sqrt{7}}{2} \\ \Rightarrow x=2\pm \sqrt{7} \end{gathered}$$