Question

Complete the square: (x-7)^2 = 8

Answer 1

Quadratic equations are all those equations that can be reformulated in a standard format (αx² + bx + c(=0) where the value of x is unknown and the values of the coefficients (a, b and c) are unknown.

These equations can always be solved using the quadratic formula, although sometimes it is also possible to use factorization or isolation of variables.

• $\large\displaystyle\begin{gathered}\sf (x-7)^{2}=8 \end{gathered}$

Expand squared

• $\large\displaystyle\begin{gathered}\sf (x-7)(x-7)=8 \end{gathered}$

It is distributed

• $\large\displaystyle\begin{gathered}\sf x(x-7)-7(x-7)=8 \end{gathered}$
• $\large\displaystyle\begin{gathered}\sf x^{2} -7x-7(x-7)=8 \end{gathered}$
• $\large\displaystyle\begin{gathered}\sf x^{2} -7x-7x+49=8 \end{gathered}$

Combine like terms.

• $\large\displaystyle\begin{gathered}\sf x^{2} -14x+49=8 \end{gathered}$

Move terms to the left

• $\large\displaystyle\begin{gathered}\sf x^{2} +14x+49-8=0 \end{gathered}$

Subtract the numbers

• $\large\displaystyle\begin{gathered}\sf x^{2} -14x+41=0 \end{gathered}$

Use the quadratic formula.

                        $\large\displaystyle\text{ \begin{gathered}\sf x=\frac{-b\pm\sqrt{b^{2}-4ac } }{2a} \end{gathered} }$

In standard form we identify "a", "b" and "c" from the original equation and add to the quadratic formula.

• a = 1
• b = -14
• c = 41

                      $\large\displaystyle\text{ \begin{gathered}\sf x=\frac{-(-14)\pm\sqrt{(-14)^{2}-4\cdot1\cdot41 } }{2\cdot1} \end{gathered} }$

Simplify

• Calculate the exponent
• multiply the numbers
• Subtract the numbers
• Calculate the square root
• multiply the numbers
• We multiply the numbers

                                         $\large\displaystyle\text{ \begin{gathered}\sf x=\frac{14\pm4\sqrt{2} }{2} \end{gathered} }$

Separate equations

To solve for the unknown variants, we split the equation into two: one with a plus sign and the other with a minus sign.

                                       $\large\displaystyle\text{ \begin{gathered}\sf x=\frac{14+4\sqrt{2} }{2} \end{gathered} }$

                                       $\large\displaystyle\text{ \begin{gathered}\sf x=\frac{14-4\sqrt{2} }{2} \end{gathered} }$

Solve

Order and isolate the variant to find each solution.

                                            $\large\displaystyle\begin{gathered}\sf x=7+2\sqrt{2} \end{gathered}\\\large\displaystyle\begin{gathered}\sf x=7-2\sqrt{2} \end{gathered}$

                            $\red{\boxed{\large\displaystyle\text{ \begin{gathered}\sf \bf{\blue{Answer \ \ \longmapsto \ \ x=7\pm2\sqrt{2} }} \end{gathered} }}}$

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Skandar