How to use the root formula
In mathematics, the root formula is an important tool for solving quadratic equations. Whether you are a student or a professional, mastering the use of root-finding formulas can help solve many practical problems. This article will introduce in detail the definition, usage and practical application examples of the root formula.
1. Definition of root formula
The root formula, also called the quadratic formula, is used to solve quadratic equations of the form ( ax^2 + bx + c = 0 ). The formula is as follows:
| formula | [ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] |
| Parameter description | a, b, c are the coefficients of the quadratic equation, and ( a neq 0 ) |
2. Steps to use the root formula
When using the root formula to solve a quadratic equation, you can follow these steps:
| Step 1 | Confirm that the equation has the form ( ax^2 + bx + c = 0 ) and determine the values of the coefficients a, b, and c. |
| Step 2 | Compute the discriminant ( D = b^2 - 4ac ). |
| Step 3 | Determine the solution of the equation based on the value of the discriminant: |
| - If ( D >0 ), the equation has two different real solutions. | |
| - If ( D = 0 ), the equation has a real solution (multiple roots). | |
| - If ( D< 0 ), the equation has no real solution, but it has a complex solution. | |
| Step 4 | Substitute a, b, and D into the root formula to find the solution to the equation. |
3. Practical application examples
Here is a concrete example showing how to use the root formula to solve a quadratic equation:
| Example | Solve the equation ( 2x^2 - 4x - 6 = 0 ). |
| Step 1 | Coefficients of determination: a = 2, b = -4, c = -6. |
| Step 2 | Calculate the discriminant: (D = (-4)^2 - 4 times 2 times (-6) = 16 + 48 = 64 ). |
| Step 3 | Discriminant ( D >0 ), the equation has two different real solutions. |
| Step 4 | Substitute into the root formula: |
| [ x = frac{-(-4) pm sqrt{64}}{2 times 2} = frac{4 pm 8}{4} ] | |
| The solution is: (x_1 = frac{4 + 8}{4} = 3), (x_2 = frac{4 - 8}{4} = -1). |
4. Precautions
When using the root formula, you need to pay attention to the following points:
| 1 | Make sure the equation is in standard quadratic form ( ax^2 + bx + c = 0 ). |
| 2 | The coefficient a cannot be 0, otherwise the equation is not quadratic. |
| 3 | The value of the discriminant ( D ) determines the properties of the solution to the equation. |
5. Summary
The root formula is a powerful tool for solving quadratic equations. You can find the solution of the equation in simple steps. Whether it is learning or practical application, it is very important to master the use of root-finding formulas. I hope the introduction in this article can help you better understand and use the root formula.
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