Algebra
Quadratic equations and functions
Solving quadratic equations by square roots
20 practice questions
2 video lessons
Theory + worked examples
Theory
The square root method solves quadratics of the form \(x^2=k\) or \((x-h)^2=k\):
\[x^2=k\ \Rightarrow\ x=\pm\sqrt{k}.\]
Remember the \(\pm\) — there are two square roots.
Take the \(\pm\) square root.
Worked examples.
The method:
\[(x-h)^2=k\ \Rightarrow\ x-h=\pm\sqrt k\]
Isolate the square before taking the root.
How to use the square root method
- Isolate the squared term.
- Take the \(\pm\) square root of both sides.
- Solve for \(x\).
- Simplify any radical.
Example 1 — Basic
Solve \(x^2=16\).
Solution
Take the square root of both sides.
| \(x\) | \(=\) | \(\pm4\) |
Example 2 — Shifted square
Solve \((x-3)^2=25\).
Solution
Take the root, then solve.
| \(x-3\) | \(=\) | \(\pm5\) |
| \(x\) | \(=\) | \(8\ \text{or}\ -2\) |
Example 3 — Isolate first
Solve \(2x^2=50\).
Solution
Divide, then take the root.
| \(x^2\) | \(=\) | \(25\) |
| \(x\) | \(=\) | \(\pm5\) |
Example 4 — Irrational root
Solve \(x^2=8\).
Solution
Simplify the radical.
| \(x\) | \(=\) | \(\pm\sqrt8=\pm2\sqrt2\) |
Common pitfalls
Include the \(\pm\) — you get two solutions.
Isolate the square first.
Simplify radicals like \(\sqrt8=2\sqrt2\).
Frequently asked questions
When can you use the square root method?
When there is no \(bx\) term (a perfect square equals a number).
Solve \(x^2=49\).
\(x=\pm7\).
Why the plus-or-minus?
Both a positive and negative number square to the same value.
Do you isolate the square first?
Yes, before taking the root.
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