How to solve equations that may be reduced to quadratic equations using substitution? Questions with detailed solutions are presented. Graphical solutions to these equations are also presented.
Question 1
Solve the equation 0.1 x4 - 1.3 x2 + 3.6 = 0 .
solution
Let u = x2 which gives u2 = x4 and rewrite the given equation in terms of u
0.1 u2 - 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 = x2 to solve for x.
u = 4 = x2 gives two solutions: x = - 2 and x = 2
u = 9 = x2 gives two solutions: x = - 3 and x = 3
The four x intercepts of the graph of y = 0.1 x4 - 1.3 x2 + 3.6 are the graphical solutions to the equation as shown below.
.
Question 2
Solve the equation: √x = 3 - (1 / 4)x.
solution
Let u = √x which gives u2= 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.
Let y = 3 - 4 / x which gives y2= (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.
Let y = (x - 1)1 / 3 which gives y2= (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 5
Find all real solutions for the following equation: \( 2\left(\dfrac{2}{x-3}\right)^2 -\dfrac{2}{x-3} - 3 = 0 \)
solution
Let u = 2 / (x - 3) which gives y2 = (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.