Q. Solve the following equation for y.2y+5=15−2yy=
Write equation: Write down the given equation.We have the equation 2y+5=15−2y.
Isolate square root term: Isolate the square root term.To solve for y, we want to isolate the square root term on one side of the equation. We can do this by subtracting 5 from both sides of the equation.2y+5−5=15−2y−5This simplifies to:2y=15−2y−5
Move y terms to one side: Move all terms involving y to one side.To make the equation easier to solve, we want all the y terms on one side. We can do this by adding 2y to both sides of the equation.2y−2y=15−2y−5+2yThis simplifies to:0=15−2y−5+2y
Rearrange equation: Rearrange the equation.We can rearrange the equation to have the square root term on one side and the other terms on the other side.15−2y=5−2y
Square both sides: Square both sides of the equation to eliminate the square root.(15−2y)2=(5−2y)2This simplifies to:15−2y=(5−2y)(5−2y)
Expand right side: Expand the right side of the equation.We need to expand the right side using the distributive property (FOIL method).15−2y=25−10y+4y2
Rearrange to quadratic equation: Rearrange the equation to form a quadratic equation.We want to bring all terms to one side to form a quadratic equation.4y2+10y−25=0
Solve quadratic equation: Solve the quadratic equation.We can solve the quadratic equation by factoring, completing the square, or using the quadratic formula. The quadratic formula is:y=2a−b±b2−4acFor our equation, a=4, b=10, and c=−25.y=2⋅4−10±102−4⋅4⋅(−25)y=8−10±100+400y=8−10±500
Simplify square root and solve: Simplify the square root and solve for y.500 can be simplified to 105.y=8−10±105We have two possible solutions for y:y=8−10+105 or y=8−10−105
Check for extraneous solutions: Check for extraneous solutions.We need to check if both solutions satisfy the original equation because squaring both sides of an equation can introduce extraneous solutions.For y=8−10+105:2y+5=15−2y2(8−10+105)+5=15−2(8−10+105)After simplifying, we check if both sides are equal.For y=8−10−105:2y+5=15−2y2(8−10−105)+5=15−2(8−10−105)After simplifying, we check if both sides are equal.If one of the solutions does not satisfy the original equation, it is an extraneous solution and should be discarded.
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