Chapter 6, Problem 8CA

a.

To determine

To solve : the quadratic equation

Answer to Problem 8CA

The solution of quadratic equation is $x=-2\pm \sqrt{14}$

Explanation of Solution

Given information :

  $x^2+4x=10$

Method used : by algebraic method

Calculation :

  $\begin{array}{l}{x}^2+4x=10\\ x^2+4x+4=10+4\\ {\left(x+2\right)}^2=14\\ x+2=\pm \sqrt{14}\\ x=-2\pm \sqrt{14}\end{array}$

Hence, the solution of quadratic equation is $x=-2\pm \sqrt{14}$ .

b.

To determine

To solve : the quadratic equation

Answer to Problem 8CA

The solution of quadratic equation is $x=2i\sqrt{3}$

Explanation of Solution

Given information :

  $x^2=-12$

Method used : by complex method

Calculation :

  $\begin{array}{l}{x}^2=-12\\ x=\sqrt{-12}\\ x=\sqrt{-3}\\ x=2\sqrt{-3}\\ x=2\sqrt{-1}\cdot \sqrt{3}\\ x=2i\sqrt{3}\end{array}$

Hence, the solution of quadratic equation is $x=2i\sqrt{3}$ .

c.

To determine

To solve : the quadratic equation

Answer to Problem 8CA

The solution of quadratic equation is $x=\frac{7\pm \sqrt{57}}{4}$

Explanation of Solution

Given information :

  $4{\left(x-1\right)}^2=6x+2$

Formula used : Sridhar Acharya

  $\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Calculation :

  $\begin{array}{l}4{\left(x-1\right)}^2=6x+2\\ 4\left(x^2-2x+1\right)=6x+2\\ 4x^2-8x+4-6x+4-2=0\\ 4x^2-8x+4-6x+4-2=0\\ 4x^2-14x-2=0\\ 2x^2-7x-1=0\\ x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\ x=\frac{-7\pm \sqrt{{\left(7\right)}^2-4\cdot 2\cdot \left(-1\right)}}{2\cdot 2}\\ x=\frac{-7\pm \sqrt{49+8}}{4}\\ x=\frac{7\pm \sqrt{57}}{4}\\ \frac{7}{4}\pm \frac{\sqrt{57}}{4}\end{array}$

Hence, the solution of quadratic equation is $x=\frac{7\pm \sqrt{57}}{4}$ .

d.

To determine

To solve : the quadratic equation

Answer to Problem 8CA

The solution of quadratic equation is $x=6,-3$

Explanation of Solution

Given information :

  $x^2-3x-18=0$

Method used : by complex method

Calculation :

  $\begin{array}{l}{x}^2-3x-18=0\\ x^2-6x+3x-18=0\\ x\left(x-6\right)+3\left(x-6\right)\\ \left(x+6\right)\left(x+3\right)=0\\ x=6,-3\end{array}$

Hence, the solution of quadratic equation is $x=6,-3$ .