Solving Radical Equations

Solving a radical equation usually involves three main steps.

Isolate the radical on one side of the equation.
Raise each side of the equation to a power equal to the index of the radical to eliminate the radical.
Solve the resulting equation.
Check the results for extraneous solutions.

Now we can analyze the given radical equation.

5 + 4 x - 5 = 12

First, let's isolate the radical,

4 x - 5,

on one side of the equation.

5 + 4 x - 5 = 12

SubEqn

LHS - 5 = RHS - 5

4 x - 5 = 7

We get an isolated radical with index equal to

2 .

Then, we will raise each side of the equation to the power of

2 .

4 x - 5 = 7

RaiseEqn

LHS^{2} = RHS^{2}

(4 x - 5)^{2} = 7^{2}

Solve for

x

PowSqrt

(a)^{2} = a

4 x - 5 = 7^{2}

CalcPowCalculate power

4 x - 5 = 49

AddEqn

LHS + 5 = RHS + 5

4 x = 54

DivEqn

LHS / 4 = RHS / 4

x = \frac{5 4}{4}

ReduceFrac

\frac{a}{b} = \frac{a / 2}{b / 2}

x = \frac{2 7}{2}

Next, we will check for extraneous solutions. We do that by substituting

\frac{2 7}{2}

for

x

into the original equation. If the substitution produces a true statement, we know that our answer is correct. If it does not, then it is an extraneous solution.

5 + 4 x - 5 = 12

Substitute

x = \frac{2 7}{2}

5 + 4 (\frac{2 7}{2}) - 5 = ? 12

Simplify

MultiplyMultiply

5 + \frac{1 0 8}{2} - 5 = ? 12

CalcQuotCalculate quotient

5 + 54 - 5 = ? 12

SubTermsSubtract terms

5 + 49 = ? 12

CalcRootCalculate root

5 + 7 = ? 12

AddTermsAdd terms

12 = 12 ✓

Because our substitution produced a true statement we know that our answer,

x = \frac{2 7}{2},

is correct.

Solving Radical Equations 1.14 - Solution