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Definition of radical equations with examples

radical equations

Radical equations (also known as irrational) are equations in which the unknown appears under a radical sign. The usual thing to do when solving these kind of equations is to put the whole equation to the power of two and solve equation without the radical.

Example 1. Solve the equation.

\sqrt{2x + 1} = 1

\sqrt{2x + 1} = 1 / ^2

2x + 1 = 1

2x = 0

x = 0

Example 2. Solve the equation.

\sqrt{2x + 1} = \sqrt{x + 2}

\sqrt{2x + 1} = \sqrt{x + 2} / ^2

2x + 1 = x + 2

x = 1

Sometimes, when squaring the whole equation, you won’t get a linear equation. If there is somewhere x that is not under the second root, you’ll get quadratic equation.

Example 3. Solve the equation.

\sqrt{x + 1} = 2x - 3

\sqrt{x + 1} = 2x - 3 / ^2

x + 1 = (2x - 3)^2

x + 1 = 4x^2 - 12x + 9

4x^2 - 13x + 8 = 0

x_1,_2 = \frac{13 \pm \sqrt{41}}{8}

There is just one more typical case that can happen. This is when you have two radical signs and a free expression. First thing, again is to square the whole equation to get rid of one root, and then leave the expression under the radical sign on the one side and transfer everything to the other side and square again.

Example 4. Solve the equation.

\sqrt{x + 1} = 2 + \sqrt{x + 2}

\sqrt{x + 1} = 2 + \sqrt{x + 2} /^2

x + 1 = 4 + 4 \sqrt{x + 2} + x + 2

4 \sqrt{x + 2} = - 5

\sqrt{x + 2} = - \frac{5}{4}

x + 2 = \frac{25}{16}

x = - \frac{7}{16}

Example 5. Solve the equation.

\sqrt{2 \sqrt{x + 1}} = 2

\sqrt{2 \sqrt{x + 1}} = 2 /^2

2 \sqrt{x + 1} = 4 /: 2

\sqrt{x + 1} = 2 /^2

x + 1 = 4

x = 3

Radical equations worksheets

  Solve radical equations (370.6 KiB, 131 hits)

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