Solving Quadratic Equations by Square Roots
Learn to solve quadratic equations by taking the square root of both sides.
Definition
- A positive root:
- A negative root:
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Worked Examples
Solve:
Identify the equation form
The equation is already in the form where → Ready to take square root
Take the square root of both sides
→
Simplify the square root
→
Write both solutions
or → Two solutions
Answer: or
Common Mistakes
Forgetting the negative solution
Why it's wrong: Students often write only the positive square root, forgetting that as well.
Correct: Always write to indicate both solutions. For , write , not just .
Taking the square root before isolating
Why it's wrong: In equations like , students sometimes try to take directly.
Correct: First divide by 2 to get , then take the square root: .
Thinking
Why it's wrong: This is only true for positive . For negative , .
Correct: Write or more precisely , which gives both positive and negative possibilities.
Attempting to solve (negative) as a real number
Why it's wrong: Students may try to write , which is not a real number.
Correct: Recognize that if , the equation has no real solution.
Interactive Visual
Balance Scale
Solution: x = 4
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Practice Problems
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Why It Matters
- Physics: Finding velocity or distance in equations like
- Geometry: Calculating side lengths from area ()
- Engineering: Determining dimensions when area is known
- Finance: Solving for rates in compound interest formulas
Real World Applications
Calculating Distances
In physics, the relationship between distance, acceleration, and time involves squared terms.
Example:
A ball is dropped and falls according to (in meters). How long does it take to fall 80 meters? Solve , so , giving seconds (we use only the positive value since time cannot be negative).
A stone is dropped from a bridge. The distance fallen is given by meters.
How long does it take to fall 45 meters?
Step 1: Write the mathematical expression
Set up the equation:
Finding Side Lengths from Area
When you know the area of a square, you can find the side length using square roots.
Example:
A square garden has an area of 144 square meters. The side length is where , so meters.
A square room has an area of 81 square meters.
What is the length of each wall?
Step 1: Write the mathematical expression
If the side length is , then
Projectile Motion
The height of a thrown ball involves quadratic equations.
Example:
A ball thrown upward reaches height feet. When is the ball at 48 feet? Solve , giving , so and second.
A fountain shoots water with height (in cm, in seconds).
At what time is the water at height 36 cm?
Step 1: Write the mathematical expression
Solve:
Key Takeaways
- 1For equations in the form , take the square root of both sides:
- 2Always include BOTH the positive and negative solutions ()
- 3First isolate before taking the square root
- 4If , there is no real solution
- 5This method also works for — just solve for the expression inside
Frequently Asked Questions
Glossary
- Square root
- A number that, when multiplied by itself, gives the original number. because .
- Perfect square
- A number that is the square of an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
- Plus-minus (±)
- A symbol indicating both positive and negative values. means both and .
- Quadratic equation
- An equation where the highest power of the variable is 2, such as or .
Formula Card
Square Root Method
Only valid when $k \geq 0$
General Form
Then solve the linear equation