In Algebra help radical equations can be defined as equation is a statement that two expressions are equal. The expression could consist with any number of terms with all combinations of numbers, variables and operators.
In certain expressions the terms may contain numbers which do not have whole number roots. Here is an algebra example, square roots of numbers which are not perfect squares, cube roots of numbers which are not perfect cubes. Such roots are called as irrational numbers which are decimals with non terminating terms. Instead of expressing those in decimal form they are conveniently expressed using radical symbols and such expressions are called as radical equations.
Radical Equations – Radical Symbols
Roots of many numbers have non terminating decimal terms.
For example, square root of 3 = 1.7320508075688772……………….
cube root of 4 = 1.587401051968199………….
It is cumber some to work with such types of numbers. But they can be comfortably referred as,
Square root of 3 as $\sqrt{3}$, cube root of 4 as $\sqrt[3]{4}$, 4th root of 7 as $\sqrt[4]{7}$ etc.
In general, the nth root of ‘a’ is denoted as $\sqrt[n]{a}$
The symbol ‘?’ is called radical symbol. The actual number and its root number are inserted as shown.
In case of square roots, the root number 2 is not inserted and hence the mere symbol ‘?’ always means a square root.
The notation also helps in basic operation operations of irrational numbers.
Examples of Radical Equations
Radical equations state that two expressions are equal, at least one expression containing radical terms, mostly square roots. The following are some simple examples in algebra
$\sqrt{x}$ = 4
$\sqrt{x}$ + 2 = 10
$\sqrt[3]{x}$ + 14 = $\sqrt{x}$ + 10
Solving Radical Expressions
A radical equation can be solved by isolating the radical terms on one side and undoing the radical terms by raising to suitable powers.
Let us take the following example
$\sqrt{x + 2}$ + 4 = x
Rewriting the equation, $\sqrt{x + 2}$ = x – 4
Squaring both sides, x + 2 = (x – 4)2 = x2 – 8x + 16
or, x2 – 9x + 14 = 0
or, (x – 7)(x – 2) = 0
So, x = 7 or x = 2
Which one of them is correct?
Check the equations with both answers.
1) $\sqrt{7 + 2}$ + 4 = 7 or $\sqrt{9}$ = 7 – 4 = 3 or $\sqrt{9}$= 3
2) $\sqrt{2 + 2}$ + 4 = 2 or $\sqrt{4}$ = 2 – 4 = -2 or $\sqrt{4}$= -2
The second check fails as the square root of a number can never be negative.
Hence the solution x = 7 is correct and the solution x = 2 is extraneous.
Always remember to check back to identify the correct solution.
In solution to equations involving even radicals you may come across extraneous solutions.
Graphing Radical Equations
The graphing radical equation gives a clearer picture to identify the correct solution.
Let us take the same example.
The graph of a square root function is always a half parabola, concave to the right. It is half parabola and lies above x- axis because the value of the function is always positive.
Graphically, the solution to the equations, y = $\sqrt{x + 2}$ and y = x -4 is the point of intersection of the graphs of the functions representing these equation.
The following diagram clearly shows that the point of intersection is only at x = 7 and hence that is the solution.
