Question

Make a substitution to write the equation in quadratic form. Then solve 2x - 9square root of x + 4 = 0

Answer 1

First, let's make the following substitutions:

`\begin{gathered} y=√(x) \\ y^2=x \end{gathered}`

Then, substituting in the given equation, we have:

`2y^2-9y+4=0`

Now, let's solve using the quadratic formula:

`\begin{gathered} a=2,b=-9,c=4\\ \\ y=(-b±√(b^2-4ac))/(2a)\\ \\ y=(9±√(81-32))/(4)\\ \\ y_1=(9+7)/(4)=4\\ \\ y_2=(9-7)/(4)=(1)/(2) \end{gathered}`

Now, let's calculate the value of x:

`\begin{gathered} y=√(x)\\ \\ \begin{cases}4=√(x)\rightarrow x=16 \\ (1)/(2)={√(x)}\rightarrow x=(1)/(4)\end{cases} \end{gathered}`

Therefore the solutions are x = 1/4 and x = 16.