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Solve the equation 0.1 x^{4}  1.3 x^{2} + 3.6 = 0 .
Solution
Let 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.
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Solve the equation: √x = 3  (1 / 4)x.
Solution
Let 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.
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Solve the equation: \( (3  \dfrac{4}{x})^2  6 (3  \dfrac{4}{x}) = 16 \)
Solution
Let y = 3  4 / x which gives y^{2}= (3  4 / x)^{2} and rewrite the given equation in terms of y.
y 2  6 y = 16
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.
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Solve the equation: 2(x  1)^{2 / 3} + 3(x  1)^{1 / 3}  2 = 0.
Solution
Let 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.
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Find all real solutions for the following equation: \( 2\left(\dfrac{2}{x3}\right)^2 \dfrac{2}{x3}  3 = 0 \)
Solution
Let 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.
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