POLQ 2 | Lesson 4 | Solutions (Explaining each Step)

Solutions: \(ax^2+bx+ c=0\)

\(x^2 +\Large\frac{b}{a}\normalsize x +\Large\frac{c}{a}= \normalsize(0)\), \(a≠0\) Dividing both sides of equation by a.
\(x^2 +\Large\frac{b}{a}\normalsize x=\Large\frac{-c}{a}\) Subtracting \(\frac{c}{a}\) from both sides
\(x^2 +\Large\frac{b}{a}\normalsize x +(\Large\frac{b}{2a})^2=\Large\frac{-c}{a}+(\Large\frac{b}{2a})^2\) Adding \((\Large\frac{b}{2a})^2\) to both sides; the square of \(\frac{1}{2}b\) to get a trinomial square.
\((x + \Large\frac{b}{2a})^2 = \Large\frac{-c}{a}+(\Large\frac{b}{2a})^2\) Factoring the left side
\((x + \Large\frac{b}{2a})^2=(\Large\frac{b}{2a})^2+\Large\frac{-c}{a}\) Commutative property of addition on the right side
\((x + \Large\frac{b}{2a})^2=(\Large\frac{b}{2a})^2+ \Large\frac{-c}{a}•\Large\frac{4a}{4a}\) To combine the fractions on the right, multiply \(\Large\frac{-c}{a}\) by \(4a\) because the LCD is \(4a^2\).
\((x + \Large\frac{b}{2a})^2= \Large\frac{b^2-4ac}{4a^2}\) Combining the fractions on the right
\(\sqrt{(x + \Large\frac{b}{2a})^2}=\sqrt {\Large\frac{b^2-4ac}{4a^2}}\) Take the square root of both sides
\(x + \Large\frac{b}{2a} = \Large\frac{±\sqrt{b^2-4ac}}{2a}\) Square root of both sides
\(x + \Large\frac{b}{2a}-\Large\frac{b}{2a} = \Large\frac{±\sqrt{b^2-4ac}}{2a}-\Large\frac{b}{2a}\) to Solve for \(x\) subtract \(\Large\frac{b}{2a}\) from both sides
\(x =\Large\frac{±\sqrt{b^2-4ac}}{2a}-\Large\frac{b}{2a}\) Simplifying the left side.
\(x =-\Large\frac{b}{2a}±\Large\frac{\sqrt{b^2-4ac}}{2a}\) Commutative property on the right side of the equation.
\(x =-\Large\frac{-b±\sqrt{b^2-4ac}}{2a}\) Combining the fractions

 

 

 


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