188k views
2 votes
I need help solving this it’s a radical equation I would appreciate the help:0

I need help solving this it’s a radical equation I would appreciate the help:0-example-1
User Metase
by
8.3k points

1 Answer

0 votes

The given expression is


\sqrt[]{6x\text{ }}+\text{ 3 = }\sqrt[]{9x\text{ + 19}}

We would square both sides of the expression. It becomes


\begin{gathered} (\sqrt[]{6x}+3)^2\text{ = (}\sqrt[]{9x+19)^2} \\ (\sqrt[]{6x}\text{ + 3)(}\sqrt[]{6x}\text{ + 3) = 9x + 19} \\ On\text{ the left, we would multiply each term in both parentheses. } \\ \sqrt[]{6x}*\sqrt[]{6x}\text{ + 3}\sqrt[]{6x}\text{ + 3}\sqrt[]{6x}\text{ + 3}*3\text{ = 9x + 19} \\ 6x\text{ + 6}\sqrt[]{6x}\text{ + 9 = 9x + 19} \\ \text{Collecting like terms, we have} \\ 6\sqrt[]{6x}\text{ = 9x - 6x + 19 - 9} \\ 6\sqrt[]{6x}\text{ = 3x + 10} \\ \text{There is still a radical on the left hand side. We would square both sides} \\ of\text{ the equation.} \\ (6\sqrt[]{6x)^2}=(3x+10)^2 \\ 6^2*6x\text{ = (3x + 10)(3x + 10)} \\ 216x=9x^2\text{ + 30x + 30x + 100} \\ 216x=9x^2+60x\text{ + 100} \\ 9x^2\text{ + 60x - 216x + 100 = 0} \\ 9x^2-\text{ 156x + 100 = 0} \end{gathered}

We have a quadratic equation. The general form of a quadratic equation is expressed as

ax^2 + bx + c = 0

By comparing our equation with the quadratic equation, we have

a = 9, b = - 156, c = - 100

We would solve for x by applying the general formula for solving quadratic equations which is expressed as


\begin{gathered} x\text{ =}\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}\text{ } \\ \text{substituting the given values, } \\ x\text{ = }\frac{-\text{ - 156 }\pm\sqrt[]{(-156)^2^{}-\text{ 4}*9\text{ x }100}}{2*9} \\ x\text{ = }\frac{156\text{ }\pm\sqrt[]{24336-\text{ 3600}}}{18} \\ x\text{ = }\frac{156\text{ }\pm\sqrt[]{20736}}{18} \\ x\text{ = }\frac{156\text{ + 144}}{18}\text{ or x = }\frac{156\text{ - 144}}{18} \\ x\text{ = }(300)/(18)\text{ or x = }(12)/(18) \end{gathered}

Dividing the numerators and denominators by 6, we have

x = 50/3 or x = 2/3

User OghmaOsiris
by
8.1k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories