Chapter 1.7, Problem 50E

Solution

To find: Correct error in the solving of the equation $x^2+10x+13=0$ .

The correct value of equation $x^2+10x+13=0$ is $x=-5\pm 2\sqrt{3}$ .

Given:

Expression is $x^2+10x+13=0$

Concept Used:

Completing square method to find the roots of the quadratic equation.

Algebraic identity: $a^2+2ab+b^2={\left(a+b\right)}^2$

Properties of square root: $a^2=b\Rightarrow a=\pm \sqrt{b}$

Calculation:

  $x^2+10x+13=0$

Applying the Completing square method of solving quadratic equation:

  $\begin{array}{l}{x}^2+10x+13=0\\ x^2+2\times 5\times x+{\left(5\right)}^2-{\left(5\right)}^2+13=0\end{array}$

Using the algebraic identity: $a^2+2ab+b^2={\left(a+b\right)}^2$ .

  $\begin{array}{c}{\left(x+5\right)}^2-25+13=0\\ {\left(x+5\right)}^2-12=0\\ {\left(x+5\right)}^2=12\\ \left(x+5\right)=\pm \sqrt{12}\\ x+5=\pm 2\sqrt{3}\left\{\because 12=2\times 2\times 3\right\}\\ x=-5\pm 2\sqrt{3}\end{array}$

Therefore, the value of $x=-5\pm 2\sqrt{3}$ .

Conclusion:

The correct value of equation is $x=-5\pm 2\sqrt{3}$ .