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Solve algebraically:

Solve algebraically:-example-1
User Kaushik NP
by
8.0k points

2 Answers

3 votes


\huge\text{Hey there!}


\huge\textbf{Equation a.}


\rm{2x^2 - 162 = 0}


\huge\textbf{Solving for:}


\rm{2x^2 - 162 = 0}


\huge\textbf{Add \boxed{\bf 162} to both sides:}


\rm{2x^2 - 162 + 162 = 0 + 162}


\huge\textbf{Simplify it:}


\rm{2x^2 = 0 + 162}


\rm{2x^2 = 162}


\huge\textbf{Divide \boxed{\bf 2} to both sides:}


\rm{(2x^2)/(2) = (162)/(2)}


\huge\textbf{Simplify it:}


\rm{x^2 = (182)/(2)}


\rm{x^2 = 81}


\huge\textbf{Take the square root of \boxed{\bf 81}}


\rm{x = \pm √(81)}


\rm{x = 9\ or\ x = -9}


\huge\textbf{Therefore, your answer should be:}


\huge\boxed{\textsf{x = } \frak{9\ or } \ \textsf{x = }\frak{-9}}\huge\checkmark


\huge\textbf{Equation b.}


\rm{-(1)/(2)(x - 3)^2 = -2}


\huge\textbf{Solving for:}


\rm{-(1)/(2)(x - 3)^2 = -2}


\huge\textbf{Simplify both sides of your equation:}


\rm{-(1)/(2)x^2 + 3x - (9)/(2) = -2}


\huge\textbf{Subtract \boxed{\bf -2} to both sides:}


\rm{-(1)/(2)x^2 + 3x - (9)/(2) - (-2) = -2 - (-2)}


\huge\textbf{Simplify it:}


\rm{- (1)/(2)x^2 + 3x - (5)/(2) = 0}


\huge\textbf{Now, we can convert the equation to:}


\rm{-0.5x^2 + 3x - 2.5 = 0}


\large\text{When we changed the equation entirely, we made it easier to solve}\uparrow


\huge\textbf{Use the quadratic formula to solve.}


\large\textsf{The quadratic formula:}


\mathsf{x= (-b \pm √(b^2 - 4ac))/(2a)}


\huge\textbf{Here are your labels:}


\text{a = }\rm{-0.5}\\\\\text{b = }\rm{ 3}\\\\\rm{c = }\rm{\ -2.5}


\huge\textbf{Your new equation:}


\rm{x = (-(3) \pm √(3^2 - 4(-0.5)(-2.5)))/(2(-0.5))}


\huge\textbf{Simplify it:}


\rm{x = (-3 \pm √(4))/(-1)}


\rm{x = 1\ or \ x = 5}


\huge\textbf{Therefore, your answer should be: }


\huge\boxed{\textsf{x = }\frak{1}\ \textsf{or x = }\frak{5}}\huge\checkmark


\huge\text{Good luck on your assignment \& enjoy your day!}

~
\frak{Amphitrite1040:)}

User Przemoc
by
7.9k points
1 vote

Answer:


a)\ x_1=-9,\ x_2=9\\\\b)\ x_1=1,\ x_2=5

Explanation:

Given equations:


a)\ 2x^2 - 162 = 0\\\\b) -(1)/(2)(x-3)^2=-2

A) 2x² - 162 = 0

Step 1: Divide both sides by 2.


\\\implies (2x^2 - 162)/(2) = (0)/(2)\\\\\implies x^2-81=0\\\\\implies x^2=81

Step 2: Take the square root of both sides (using both the positive and negative roots).


\\\implies √(x^2)=√(81)\\\\\implies x=\pm\ 9

Step 3: Separate into two cases.


\implies x_1 = -9,\ x_2 =-9

----------------------------------------------------------------------------------------------------------------

B) -1/2(x - 3)² = -2

Step 1: Multiply both sides by -2.


\\\implies -2\left(-(1)/(2)(x-3)^2\right)=-2(-2)\\\\\implies (x-3)^2=4

Step 2: Take the square root of both sides (using both the positive and negative roots).


\\\implies√((x-3)^2)=√(4)\\\\\implies x-3=\pm\ 2

Step 3: Separate into two cases and solve each one.
1)\ x-3=-2\implies x=-2+3\implies \boxed{x=1}\\\\2)\ x-3=2\implies x=2+3\implies \boxed{x=5}

User Albin Sunnanbo
by
9.4k points

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