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.
[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]
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Skandar
