Question

Create a list of steps, in order, that will solve the following equation.1/2(x+5)^2+2=42.5

Answer 1

Removing fractions, taking squared expression on one side, removing squares and getting value of x can be the major steps for needed solution.

How to find the roots of a quadratic equation?

Suppose that the given quadratic equation is

$ax^2 + bx + c = 0$

Then its roots are given as:

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

How to find the factors of a quadratic expression?

If the given quadratic expression is of the form $ax^2 + bx + c = 0$

then its factored form is obtained by two numbers alpha( α ) and beta( β) such that:

$b = \alpha + \beta \\ ac =\alpha \times \beta$

Then writing b in terms of alpha and beta would help us getting common factors out.

Sometimes, it is not possible to find factors easily, so using the quadratic equation formula can help out without any trial and error.

For this case, we're given the equation

$\dfrac{1}{2}(9x+5)^2 + 2 = 42.5$

One good thing of this equation is that the variable x is already in linear form inside that bracket which is square. We can use operations on both the sides to remove that square and then get the value of the x which will be the solution of the considered equation.

Step 1: Removing fractions

$\dfrac{1}{2}(x+5)^2 + 2 = 42.5\\\\(x+5)^2 + 4 = 85$

Step 2: Taking squared expression on one side, and rest on other side:

$(x+5)^2 + 4 = 85\\\\\text{subtracting 4 from both the sides}\\\\(x+5)^2 = 81$

Step 3: Taking roots to remove square:

$\sqrt{(x+5)^2} = \sqrt{(\pm9)^2}\\\\x+5 = \pm 9$ (it is because 81 can be rewritten as square of 9 or -9)

That gives us two possible equations as:

$x+5 = 9\\or\\x+5 = -9$

From first equation, we get:

x = 9-5 = 4

From second equation, we get:

x + 5 = -9

x = -9-5 = -14

x=4 or -14 will make that equation true.

Thus, removing fractions, taking squared expression on one side, removing squares and getting value of x can be the major steps for needed solution.

Learn more about finding the solutions of a quadratic equation here:

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