What this move does
An equation with the unknown under a square root is called a radical equation. To get rid of the root, square both sides:
- Get the square root on its own on one side. (Otherwise the root is still there after squaring.)
- Square both sides. Square the right side as a whole, using the perfect square formula.
- Solve the polynomial equation you get.
- Substitute back into the original equation to check. Any root that makes the right side negative is an extraneous root and must be rejected.
Why do extraneous roots appear? Because squaring “erases” the sign: , and is also . So the two different equations below become the same equation once squared:
In the picture, is the upper half of a parabola and is the lower half. After squaring, every point where the line meets the whole parabola counts as a “root”. The lower half is not part of the original equation at all; we call it the ghost branch. An intersection on the ghost is an extraneous root.
Predict first
How many real solutions does have?
A square root like is never negative, so it cannot equal . You know without calculating anything: no solution.
If you square anyway, you get , so . But substitute it back: the left side is and the right side is . In the picture below, the horizontal line is below the axis, so it can only meet the ghost branch.
Work through one
Example: solve √(2x+7) − x = −4
Step 1 / 6- Get the square root alone on the left
Move to the right: . In the picture, the solid curve is and the line is . Their intersections are what we want.
Let the line slide down
The number of solutions depends on how many times the line meets the upper half. Slowly move the line down (make bigger): intersections appear, separate, and then one of them slips onto the ghost.
Predict first
Take the equation . As increases from to , how does the number of real solutions change?
Drag and watch:
- : the line is above the curve with no intersections: no solution.
- : tangent, one solution.
- : both intersections are on the upper half: two solutions.
- : one intersection has slid onto the ghost branch, leaving one solution.
The squared equation still has two roots when . Look only at that, and you would count one too many.
The same thing keeps appearing under the root? Substitute
Example: solve x² + 3x + √(x²+3x) = 6
Step 1 / 4- Find the piece that keeps coming back
appears twice: once outside the root and once inside it. Let . Then , and .
When things go wrong
Break it
Counterexample lab: remove one condition
Conclusion: the squared equation has exactly the same solutions as the original ← still true?
: squaring gives , so or . Both values make the right side . Check them one by one: both are genuine roots.
Change only the sign of the right side: . After squaring it is still , and the candidates are still and . Click each one and see which gives itself away.
At the right side is , which is negative, so it has become an extraneous root. Two different equations turn into the same equation when squared, so the squared equation cannot tell you which root belongs to which. Only substituting back into the original can.
Practice
Solve .
Solve .
Solve .
Solve .