Question

Solve using formula.
(1) x² + 6x + 5 = 0
(2) x² – 3x – 2 = 0
(3) 3m² + 2m – 7 = 0
(4) 5m² – 4m – 2 = 0
(5) $y^2+\frac{1}{3}y=2$
(6) 5x² + 13x + 8 = 0

Answer 1

(1) x² + 6x + 5 = 0
On comparing with the equation $a{x}^2+bx+c=0$,
a = 1, b = 6 and c = 5
Now $b^2-4ac=6^2-4\times 1\times 5=36-20=16$
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
$$\begin{gathered} x=\frac{-6\pm \sqrt{16}}{2\times 1}=\frac{-6\pm 4}{2} \\ \Rightarrow x=\frac{-6+4}{2}\mathrm{or}x=\frac{-6-4}{2} \\ \Rightarrow x=-1\mathrm{or}x=-5 \end{gathered}$$

(2) x² – 3x – 2 = 0
On comparing with the equation $a{x}^2+bx+c=0$,
a = 1, b = $-$3 and c = $-$2
Now $b^2-4ac={\left(-3\right)}^2-4\times 1\times \left(-2\right)=9+8=17$
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
$$\begin{gathered} x=\frac{3\pm \sqrt{17}}{2\times 1}=\frac{3\pm \sqrt{17}}{2} \\ \Rightarrow x=\frac{3+\sqrt{17}}{2}\mathrm{or}x=\frac{3-\sqrt{17}}{2} \end{gathered}$$

(3) 3m² + 2m – 7 = 0
On comparing with the equation $a{x}^2+bx+c=0$,
a = 3, b = 2 and c = $-$7
Now $b^2-4ac={\left(2\right)}^2-4\times 3\times \left(-7\right)=4+84=88$
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
$$\begin{gathered} x=\frac{-2\pm \sqrt{88}}{2\times 3}=\frac{-2\pm \sqrt{88}}{6} \\ \Rightarrow x=\frac{-2+\sqrt{88}}{6}\mathrm{or}x=\frac{-2-\sqrt{88}}{6} \\ \Rightarrow x=\frac{-1+\sqrt{22}}{3}\mathrm{or}\frac{-1-\sqrt{22}}{3} \end{gathered}$$

(4) 5m² – 4m – 2 = 0
On comparing with the equation $a{x}^2+bx+c=0$,
a = 5, b = – 4and c = $-$2
Now $b^2-4ac={\left(-4\right)}^2-4\times 5\times \left(-2\right)=16+40=56$
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
$$\begin{gathered} x=\frac{-4\pm \sqrt{56}}{2\times 5}=\frac{-4\pm \sqrt{56}}{10} \\ \Rightarrow x=\frac{-4+\sqrt{56}}{10}\mathrm{or}x=\frac{-4-\sqrt{56}}{10} \\ \Rightarrow x=\frac{-2+\sqrt{14}}{5}\mathrm{or}\frac{-2-\sqrt{14}}{5} \end{gathered}$$

(5) $y^2+\frac{1}{3}y=2$
Multiplying the equation by 3
$$\begin{gathered} 3y^2+y=6 \\ \Rightarrow 3y^2+y-6=0 \end{gathered}$$
On comparing with the equation $a{x}^2+bx+c=0$,
a = 3, b = 1 and c = $-$6
Now $b^2-4ac={\left(1\right)}^2-4\times 3\times \left(-6\right)=1+72=73$
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
$$\begin{gathered} x=\frac{-1\pm \sqrt{73}}{2\times 3}=\frac{-1\pm \sqrt{73}}{6} \\ \Rightarrow x=\frac{-1+\sqrt{73}}{6}\mathrm{or}x=\frac{-1-\sqrt{73}}{6} \end{gathered}$$

(6) 5x² + 13x + 8 = 0
On comparing with the equation $a{x}^2+bx+c=0$,
a = 5, b = 13 and c = 8
Now $b^2-4ac={\left(13\right)}^2-4\times 5\times 8=169-160=9$
$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$
$$\begin{gathered} x=\frac{-13\pm \sqrt{9}}{2\times 5}=\frac{-13\pm 3}{10} \\ \Rightarrow x=\frac{-13+3}{10}\mathrm{or}x=\frac{-13-3}{10} \\ \Rightarrow x=\frac{-10}{10}\mathrm{or}x=\frac{-16}{10} \\ \Rightarrow x=-1\mathrm{or}x=\frac{-8}{5} \end{gathered}$$