Quadratic by Taking Square Roots Calculator
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Reviewed by Calculator Editorial Team
This calculator solves quadratic equations using the square root method. It's perfect for students, engineers, and anyone needing to find the roots of quadratic equations in the form ax² + bx + c = 0.
How to Use This Calculator
To solve a quadratic equation using the square root method:
- Enter the coefficients a, b, and c from your quadratic equation in the form ax² + bx + c = 0
- Click "Calculate" to find the roots
- Review the results and interpretation
- Use the reset button to clear the calculator for a new equation
The square root method works best when the quadratic equation has real roots. For complex roots, consider using the quadratic formula instead.
The Formula Explained
The square root method for solving quadratic equations is based on completing the square. The standard form is:
ax² + bx + c = 0
The solution involves:
- Dividing the equation by a to make the coefficient of x² equal to 1
- Moving the constant term to the other side
- Completing the square by adding (b/2a)² to both sides
- Taking the square root of both sides
- Solving for x
The roots are given by:
x = [-b ± √(b² - 4ac)] / (2a)
Worked Examples
Example 1: Simple Quadratic Equation
Solve x² + 5x + 6 = 0
| Step | Calculation |
|---|---|
| 1 | a = 1, b = 5, c = 6 |
| 2 | Discriminant = b² - 4ac = 25 - 24 = 1 |
| 3 | Roots: x = [-5 ± √1]/2 |
| 4 | x₁ = (-5 + 1)/2 = -2 |
| 5 | x₂ = (-5 - 1)/2 = -3 |
Example 2: Quadratic with Fractional Coefficients
Solve 2x² - 4x - 6 = 0
| Step | Calculation |
|---|---|
| 1 | a = 2, b = -4, c = -6 |
| 2 | Discriminant = b² - 4ac = 16 - (-48) = 64 |
| 3 | Roots: x = [4 ± √64]/4 |
| 4 | x₁ = (4 + 8)/4 = 3 |
| 5 | x₂ = (4 - 8)/4 = -1 |
Frequently Asked Questions
- What is the square root method for quadratic equations?
- The square root method is an algebraic technique for solving quadratic equations by completing the square and then taking the square root of both sides.
- When should I use the square root method?
- Use the square root method when the quadratic equation has real roots and you want to solve it by completing the square. It's particularly useful when the equation is already close to a perfect square.
- What if the discriminant is negative?
- If the discriminant (b² - 4ac) is negative, the equation has complex roots. In this case, you should use the quadratic formula which can handle complex numbers.
- Can this calculator solve any quadratic equation?
- Yes, this calculator can solve any quadratic equation in the form ax² + bx + c = 0, whether it has real or complex roots.
- How accurate are the results?
- The calculator uses standard floating-point arithmetic, so results are accurate to about 15 decimal places. For more precise calculations, consider using symbolic computation software.