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Wize Grade 11 Mathematics Textbook > Radical Expressions and Functions (Content Coming Soon)

Solving Radical Equations

Operations with RadicalsPrevious Section
Radical FunctionsNext Section

Solving Radical Equations



To solve an equation involving a radical we want to
  1. Isolate the radical on one side of the equal sign
  2. Raise both sides of the equation to a power that matches the index
  3. Simplify and solve as normal
  4. Check solutions in the original equation

Wize Tip
If there is more than one radical that need to be eliminated, you will have repeat the steps that isolate it, and raise it to a power.


Example 1
Solve the following equations

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

2x+3=x+2(2x+3)2=(x+2)22x+3=x2+4x+40=x2+2x+10=(x+1)(x+1)0=(x+1)2x=−1\begin{aligned} \sqrt{2x + 3} &= x + 2 \\ (\sqrt{2x+3})^2 &= (x+2)^2 \\ 2x + 3 &= x^2 + 4x + 4 \\ 0 &= x^2 + 2x + 1 \\ 0 &= (x + 1)(x+1) \\ 0 &= (x + 1)^2 \\ x &= -1 \\ \end{aligned}

2. 4x−2−3x+1=0\sqrt{4x - 2} - \sqrt{3x + 1} = 0

4x−2−3x+1=04x−2=3x+1(4x−2)2=(3x+1)24x−2=3x+1x−2=1x=3\begin{aligned} \sqrt{4x - 2} - \sqrt{3x + 1} &= 0 \\ \sqrt{4x - 2} &= \sqrt{3x +1} \\ (\sqrt{4x - 2})^2 &= (\sqrt{3x + 1})^2 \\ 4x - 2 &= 3x + 1 \\ x -2 &= 1 \\ x &= 3 \end{aligned}

Example: Solving Radical Equations

Mia was working on building a fence around a triangular portion of their garden.
If she has exactly 12 meters of fencing, how the missing sides of the garden?


First we want to set up an equations that represents the fencing Mia used

12=3+x−22+x−112=x−19+x−1\begin{aligned} 12 &= 3 + x - 22 + \sqrt{x-1} \\ 12 &= x - 19 + \sqrt{x-1} \end{aligned}
To find the missing sides we have to solve this equation for xx.

12=x−19+x−131−x=x−1(31−x)2=(x−1)2961−62x+x2=x−1x2−63x+962=0(x−26)(x−37)=0\begin{aligned} 12 &= x - 19 + \sqrt{x - 1} \\ 31 - x &= \sqrt{x - 1} \\ (31 - x)^2 &= (\sqrt{x-1})^2 \\ 961 - 62x + x^2 &= x - 1 \\ x^2 - 63x + 962 &= 0 \\ (x - 26)(x - 37) & = 0 \end{aligned}

So we have that x=26x = 26 or x=37x = 37.

If x=26x = 26 then the sides of the triangle will be 3, 4, and 5 meters.
The agrees with the fact that Mia has 12 meters of fencing.

If x=37x = 37 then the sides of the triangle will be 3, 15, 6 meters.
The would mean Mia had 24 meters of fencing, which does not agree with the given information.


checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.

Extra Practice
Solving Radical Equations
Solve for a:
2a+4−3=3a−22\sqrt{a+4}-3=\sqrt{3a-2}
Solving Radical Equations
Solve for xx:
2x−1=x\sqrt{2x-1}=\sqrt{x}
Solving Radical Equations
Practice: Solving Radical Equations
Solve algebraically, stating any restrictions:
x−6+3=x+9\sqrt{x-6}+3=\sqrt{x+9}
Solving Radical Equations
Practice: Solving Radical Equations
Solve algebraically, stating any restrictions:
x2−2=x+4\sqrt{x^2-2}=\sqrt{x+4}
Solving Radical Equations
Practice: Solving Radical Equations
Solve algebraically, stating any restrictions on xx:
10+10x−1=13\begin{array}{rcl} 10+\sqrt{10x-1}&=&13 \end{array}
Solving Radical Equations
Practice: Solving Radical Equations
Solve algebraically, stating any restrictions on xx:
x−3=3x−4\begin{array}{rcl} x-3&=&3\sqrt{x-4} \end{array}