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more_equations - More Equations Disguised quadratic...

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More Equations Disguised quadratic equations Example 1 : Solve for x : = + x 4 3 4 x 2 . Solution : = + x 4 3 4 x 2 ------- (i). is equivalent to the equation = - + x 4 4 x 2 3 0. Letting = u x 2 we obtain a quadratic equation for u . = - + u 2 4 u 3 0. The left-hand side can be factored to give = ( ) - u 1 ( ) - u 3 0. This gives = u 1 or = u 3. Writing these two equations in terms of x gives: = x 2 1 or = x 2 3. Solving each of these equations for x gives = x - 1 or = x 1 or = x - 3 or = x 3. Hence the solution set of equation (i) is { } , , , - 3 - 1 1 3 . _______ Note : The solutions of equation (i) give the x coordinates of the points of intersection of the two graphs = y + x 4 3 and = y 4 x 2 . The solutions can also be visualised as the x intercepts of the graph of = y - + x 4 4 x 2 3.
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Example 2 : Solve for x : = - + 2 x 9 x 4 0. Solution : = - + 2 x 9 x 4 0 ------- (ii). Letting = r x we obtain a quadratic equation for r . = - + 2 r 2 9 r 4 0. The left-hand side can be factored to give = ( ) - 2 r 1 ( ) - r 4 0. This gives = r 1 2 or = r 4. Writing these two equations in terms of x gives: = x 1 2 or = x 4. Solving each of these equations for x gives = x 1 4 or = x 16. Hence the solution set of equation (i) is { } , 1 4 16 . ____ Notes : (1) The solutions can be checked in the original equation (ii). When = x 1 4 , the left-hand side of (ii) is: = - + 2 1 4 9 1 4 4 - + 1 2 9 2 4 = 0. When = x 16, the left-hand side of (ii) is: = - + 2 ( ) 16 9 16 4 - + 32 36 4 = 0. (2) Since (ii) is equivalent to the equation = + 2 x 4 9 x , the solutions of (ii) can also be visualised as the x coordinates of the points of interection of the line = y + 2 x 4 with the curve = y 9 x .
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(3) The solutions can also be visualised as the x intercepts of the graph of = y - + 2 x 9 x 4. (4) The equation (ii) can be solved in another way. Isolating the term involving the square root on the right side of the equation gives: = + 2 x 4 9 x . Squaring both sides of this equation gives: = + + 4 x 2 16 x 16 81 x . This is a quadratic equation for x and it can be written in the form: = - + 4 x 2 65 x 16 0. The left-hand side can be factored to give the equivalent equation: = ( ) - 4 x 1 ( ) - x 16 0, so gives: = x 1 4 or = x 16, as before. The process of squaring an equation can introduce extraneous solutions , but we this has not happened here since we know that both of the values = x 1 4 and = x 16 check in the original equation.
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