How to solve equations that may be reduced to quadratic equations using substitution? Examples with detailed solutions are presented. Graphical solutions to these equations are also presented.
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Question 1Solve the equation 0.1 x^{4}  1.3 x^{2} + 3.6 = 0 .solutionLet u = x^{2} which gives u^{2} = x^{4} and rewrite the given equation in terms of u 0.1 u^{2}  1.3 u + 3.6 = 0 Solve the above quadratic equation to find u. u = 4 and u = 9 We now use the substitution u = x^{2} to solve for x. u = 4 = x^{2} gives two solutions: x =  2 and x = 2 u = 9 = x^{2} gives two solutions: x =  3 and x = 3 The four x intercepts of the graph of y = 0.1 x^{4}  1.3 x^{2} + 3.6 are the graphical solutions to the equation as shown below. .
Question 2Solve the equation: √x = 3  (1 / 4)x.solutionLet u = √x which gives u^{2}= x and rewrite the given equation in terms of u u = 3  u ^{2} / 4 Multiply all terms by 4, simplify and write the above quadratic in standard form and solve it for u. u ^{2} + 4 u  12 = 0 Two solutions: u =  6 and u = 2 Use the substitution used above u = √x to solve for x. u =  6 = √x has no solution u = 2 = √x has solution x = 4 Below is shown the graph of the right side of the given equation when written with its right side equal to zero. The x intercept of the graph is the graphical solution to the equation as shown below. .
Question 3Solve the equation: \( (3  \dfrac{4}{x})^2  6 (3  \dfrac{4}{x}) = 16 \)solutionLet y = 3  4 / x which gives y^{2}= (3  4 / x)^{2} and rewrite the given equation in terms of y. y Solve the above equation. y ^{2}  6 y  16 = 0 y =  2 and y = 8 y =  2 and y = 8 Solve for x. First solution: y = 3  4 / x = 2 gives x = 4 / 5 First solution: y = 3  4 / x = 8 gives x =  4 / 5 The graph of the right side of the given equation written with its right side equal to zero. The x intercepts of the graph are the graphical solutions to the equation as shown below. .
Question 4Solve the equation: 2(x  1)^{2 / 3} + 3(x  1)^{1 / 3}  2 = 0.solutionLet y = (x  1)^{1 / 3} which gives y^{2}= (x  1)^{2 / 3} and rewrite the given equation in terms of y. 2 y ^{2} + 3 y  2 = 0 y =  2 and y = 1 / 2 Solve for x. y = (x  1)^{1 / 3} =  2 gives x = 7 y = (x  1)^{1 / 3} = 1 / 2 gives x = 9 / 8 The graph of the right side of the given equation is shown below and its x intercepts are the graphical solutions to the given equation. .
Question 5Find all real solutions for the following equation: \( 2\left(\dfrac{2}{x3}\right)^2 \dfrac{2}{x3}  3 = 0 \)solutionLet u = 2 / (x  3) which gives y^{2} = (2 / (x  3))^{2} and rewrite the given equation in terms of u. 2 u ^{2}  u  3 = 0 Solve for u. u =  1 and u = 3 / 2 Solve for x. y = 2 / (x  3) =  1 gives x = 1 y = 2 / (x  3) = 3 / 2 gives x = 13 / 3 Below is shown the graph of the right side of the equation and its x intercepts which are the graphical solution to the given equation. .
