Question

Solve the equation by identifying the quadratic form. Use a substitute variable(t) and find all real solutions by factoring. Type your answers from smallest to largest. If an answer is not an integer then type it as a decimal rounded to the nearest hundredth. When typing exponents do not use spaces and use the carrot key ^ (press shift and 6). For example, x cubed can be typed as x^3.4(x-1)^2-9(x-1)=-2Step 1. Identify the quadratic formLet t= Answer. We now have:4t^2-9t=-2Step 2. FactorFactor this and solve for t to get t=Answer and Answer Step 3. Solve for xWe have solved for t now we need to use this value for t to help us solve for x. Revisit step 1 to remind you of the relationship between t and x. Type your answers from smallest to largest.x=Answer and Answer

Answer 1

You have the following expression:

`4(x-1)^2-9(x-1)=-2`

If t = x - 1, then, the previous expression can be written as follow:

`\begin{gathered} 4t^2-9t=-2 \\ 4t^2-9t+2=0 \end{gathered}`

That is, you obtain a quadratic equation for t, Use the quadratic formula to find the values of t, as follow:

`\begin{gathered} t=\frac{-(-9)\pm\sqrt[]{(-9)^2-4(4)(2)}}{2(4)} \\ t=\frac{9\pm\sqrt[]{49}}{8} \\ t_1=(9+7)/(8)=(16)/(8)=2 \\ t_2=(9-7)/(8)=(2)/(8)=(1)/(4)=0.25 \end{gathered}`

Hence, the solutions for t are:

t = 2

t = 0.25

Now, use the previous result for t into the expression t = x - 1, to find the values of x, as follow:

t = x - 1

x = t + 1

x = 2 + 1 = 3

x = 0.25 + 1 = 1.25

Hence, the solution to the initiale quation are:

x = 1.25

x = 3