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

Sagot :

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.

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

Expand squared

  • [tex]\large\displaystyle\text{$\begin{gathered}\sf (x-7)(x-7)=8 \end{gathered}$}[/tex]

It is distributed

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

Combine like terms.

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

Move terms to the left

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

Subtract the numbers

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

Use the quadratic formula.

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

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

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

Simplify

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

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

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.

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

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

Solve

Order and isolate the variant to find each solution.

                                            [tex]\large\displaystyle\text{$\begin{gathered}\sf x=7+2\sqrt{2} \end{gathered}$}\\\large\displaystyle\text{$\begin{gathered}\sf x=7-2\sqrt{2} \end{gathered}$}[/tex]

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

Learn more at:https://brainly.com/question/22842654

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