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.
Understanding Math: Solving Quadratics by Square Roots
The main skill is not the square root step itself. It is preparing the equation so that one squared expression stands alone. Start by using inverse operations.
Add or subtract to move ordinary number terms away from the squared term. Then divide if a coefficient is multiplying the square. For example, if three times x squared minus twelve equals zero, add twelve first.
This gives three times x squared equals twelve. Divide both sides by three before taking roots.
The result is x squared equals four, which leads to two values for x. Taking a root before isolating the square is a common source of errors.
A coefficient can change the roots in an important way. Consider four times x squared equals twenty five. Dividing by four gives x squared equals twenty five fourths.
The square root of twenty five fourths is five halves. The answers are positive five halves and negative five halves. Students sometimes divide the answer by four instead of taking the root of the fraction.
That does not undo the original multiplication correctly. Another useful habit is simplifying roots when possible.
The square root of seventy two can be written as six times the square root of two because seventy two contains thirty six as a square factor. This form is exact, while a decimal is only an approximation.
Expressions with parentheses need careful attention. If the equation says the quantity x plus three, all squared, equals sixteen, first take the roots of sixteen. This produces x plus three equals four or x plus three equals negative four.
Only then should three be subtracted from both cases. The answers become one and negative seven. The number inside the parentheses shifts the graph left or right, so it changes the final answers in the opposite direction from its displayed sign.
A plus three inside the square means subtract three later. Writing each line clearly helps prevent a sign mistake.
This method connects directly to graphs and measurement. A graph of a squared expression is a parabola. Solving by square roots finds the horizontal positions where that parabola reaches a chosen height.
For a positive value on the right side, there are often two positions because a parabola has matching left and right sides. In science, squared quantities appear in area, distance formulas, and motion relationships. A square root can therefore represent a length, speed, or time value.
Context may rule out one algebraic answer. A negative length is not meaningful, even though it can solve an equation before units are considered. Keep exact roots until the final step, check both results in the original equation, and make sure any answer fits the situation.
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.