Question

How do you solve an equation by completing the square?

Answer 1

Explanation for solving equation by completing the square:

• Let a quadratic equation be ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$
• Then write the equation in the form such that $\mathrm{c}$ is on the right side i.e. ${\mathrm{ax}}^2+\mathrm{bx}=-c$.
• Divide the complete equation by $\mathrm{a}$, if $\mathrm{a}\ne 1$, then equation becomes ${\mathrm{x}}^2+\frac{\mathrm{b}}{\mathrm{a}}\mathrm{x}=-\frac{\mathrm{c}}{\mathrm{a}}$.
• Add the square of the half of the coefficient of the term $\text{'}\mathrm{x}\text{'}$i.e., ${\left(\frac{\mathrm{b}}{2\mathrm{a}}\right)}^2$on both sides, the equation becomes ${\mathrm{x}}^2+\frac{\mathrm{b}}{\mathrm{a}}\mathrm{x}+{\left(\frac{\mathrm{b}}{2\mathrm{a}}\right)}^2=-\frac{\mathrm{c}}{\mathrm{a}}+{\left(\frac{\mathrm{b}}{2\mathrm{a}}\right)}^2$
• Apply algebraic identity $p^2+2pq+q^2={\left(p+q\right)}^2$, i.e., ${\left(\mathrm{x}+\frac{\mathrm{b}}{2\mathrm{a}}\right)}^2=-\frac{\mathrm{c}}{\mathrm{a}}+{\left(\frac{\mathrm{b}}{2\mathrm{a}}\right)}^2$
• Take the square root on both sides, i.e., $\left(\mathrm{x}+\frac{\mathrm{b}}{2\mathrm{a}}\right)=\pm \sqrt{-\frac{\mathrm{c}}{\mathrm{a}}+{\left(\frac{\mathrm{b}}{2\mathrm{a}}\right)}^2}$
• Solve for variable $\text{'}\mathrm{x}\text{'}$ and find the roots, i.e., $\mathrm{x}=-\frac{\mathrm{b}}{2\mathrm{a}}\pm \sqrt{\frac{b^2-4ac}{4a^2}}\Rightarrow x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Hence, solving an equation by completing the square method is described above.