Some quadratic equations can be solved quickly by taking square roots. This works especially well when the equation can be written in the form x^2 = k or (x - h)^2 = k. The key idea is that squaring hides the sign of a number, so undoing a square usually gives two possible answers.
The plus-or-minus symbol is the reminder that both positive and negative roots may work.
When you take the square root of both sides, you are finding all numbers whose square gives the same value. If k is positive, x^2 = k has two real solutions: x = sqrt(k) and x = -sqrt(k). If k = 0, there is only one real solution, x = 0, and if k is negative, there are no real solutions unless complex numbers are allowed.
This method is most useful when there is no x term, or when the quadratic is already written as a perfect square.
Key Facts
- If x^2 = k, then x = ±sqrt(k).
- If k > 0, x^2 = k has two real solutions.
- If k = 0, x^2 = 0 has one real solution: x = 0.
- If k < 0, x^2 = k has no real solutions.
- If (x - h)^2 = k, then x - h = ±sqrt(k), so x = h ± sqrt(k).
- Always check solutions by substituting them back into the original equation.
Vocabulary
- Quadratic equation
- An equation involving a squared variable, often written in a form related to ax^2 + bx + c = 0.
- Square root
- A number that gives a specified value when multiplied by itself.
- Plus-or-minus
- The symbol ± means that both the positive and negative values should be considered.
- Perfect square
- An expression that can be written as something multiplied by itself, such as x^2 or (x - 3)^2.
- Real solution
- A solution that is a real number and can be placed on the number line.
Common Mistakes to Avoid
- Forgetting the negative root: Writing x = sqrt(k) for x^2 = k misses the fact that both sqrt(k) and -sqrt(k) square to k.
- Taking the square root before isolating the square: In 3x^2 = 48, you must first divide by 3 to get x^2 = 16, then take square roots.
- Using ± when k = 0 as if there are two answers: The values +0 and -0 are the same number, so x^2 = 0 has only one solution.
- Trying to use this method on every quadratic: Equations like x^2 + 5x = 6 are not ready for square roots because the x term prevents direct isolation of a perfect square.
Practice Questions
- 1 Solve x^2 = 49.
- 2 Solve 2(x - 3)^2 = 50.
- 3 Explain why x^2 = -9 has no real solutions, but x^2 = 9 has two real solutions.