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How to find a quadratic equation?

Published in Quadratic Equations 3 mins read

Finding a quadratic equation depends on the information you have. Here are a few common scenarios and how to address them:

1. Finding a Quadratic Equation Given its Roots

If you know the roots (solutions) of the quadratic equation, let's call them r1 and r2, you can construct the quadratic equation as follows:

Method:

  1. Form the factors: (x - r1) and (x - r2)
  2. Multiply the factors: (x - r1)(x - r2) = 0
  3. Expand the equation: x2 - (r1 + r2)x + r1 r2 = 0

Example:

Let the roots be 2 and -3.

  1. Factors: (x - 2) and (x + 3)
  2. Equation: (x - 2)(x + 3) = 0
  3. Expanded: x2 + x - 6 = 0

2. Solving a Quadratic Equation: Using the Quadratic Formula

If you already have a quadratic equation in the form ax2 + bx + c = 0, you can find the roots using the quadratic formula.

The Quadratic Formula:

x = [-b ± √(b2 - 4ac)]/2a

Explanation:

  • a, b, and c: These are the coefficients of the quadratic equation ax2 + bx + c = 0.
  • ±: The plus-minus symbol indicates that there are two possible solutions for x: one using addition and one using subtraction.
  • √(b2 - 4ac): This part is called the discriminant. It determines the nature of the roots (real, distinct, real and equal, or complex).

Steps to Apply the Quadratic Formula:

  1. Identify a, b, and c: From your equation ax2 + bx + c = 0.
  2. Substitute: Plug the values of a, b, and c into the quadratic formula.
  3. Simplify: Calculate the expression to find the two values of x. One solution will be [-b + √(b2 - 4ac)]/2a, and the other will be [-b - √(b2 - 4ac)]/2a.

Example:

Solve the equation 2x2 + 5x - 3 = 0

  1. a = 2, b = 5, c = -3
  2. x = [-5 ± √(52 - 4 2 -3)] / (2 * 2)
  3. x = [-5 ± √(25 + 24)] / 4
  4. x = [-5 ± √49] / 4
  5. x = [-5 ± 7] / 4
  6. x1 = (-5 + 7) / 4 = 2/4 = 1/2
  7. x2 = (-5 - 7) / 4 = -12/4 = -3

Therefore, the solutions are x = 1/2 and x = -3.

3. Constructing a Quadratic Equation From the Sum and Product of Roots

If you know the sum (S) and product (P) of the roots, the quadratic equation can be written as:

x2 - Sx + P = 0

Example:

If the sum of the roots is 4 and the product is 3, the equation is:

x2 - 4x + 3 = 0

Method Information Needed Result
Roots (r1, r2) The values of the roots x2 - (r1 + r2)x + r1 * r2 = 0
Quadratic Formula (ax2 + bx + c = 0) Coefficients a, b, and c x = [-b ± √(b2 - 4ac)]/2a
Sum (S) and Product (P) of Roots The sum and product of the equation's roots x2 - Sx + P = 0

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