Understanding Quadratic Equations and their Roots

A quadratic equation is an algebraic expression of the form ax2 bx c 0, where a, b, and c are constants, and a ≠ 0. The roots of the quadratic equation are the values of x that satisfy the equation. We can find the quadratic equation given its roots, using the fact that if r1 and r2 are the roots of the quadratic equation, the equation can be expressed as:

Steps to Form the Quadratic Equation

Step 1: Identify the Roots

Given the roots of the quadratic equation are -6 and 10, we can use the following formula to derive the equation:

Forming the Quadratic Equation from Given Roots

If r1 and r2 are the roots, the quadratic equation can be written as:

x2 - (r1 r2)x (r1 * r2) 0

Substituting the given values r1 -6 and r2 10 into this equation:

S r1 r2 -6 10 4

P r1 * r2 -6 * 10 -60

Therefore, the quadratic equation is:

Deriving the Final Quadratic Equation

The quadratic equation for the given roots -6 and 10 is then formed as:

x2 - Sx P 0

Substituting the calculated values:

x2 - 4x - 60 0

We can verify this using the fact that the roots of the equation x2 - 4x - 60 0 can be found by factoring:

x2 - 4x - 60 (x 6)(x - 10) 0

This confirms that the roots are -6 and 10.

Factoring Method

Alternatively, we can factorize the quadratic equation:

A(x - r1) (x - r2) 0

Substituting the roots:

x2 - (r1 r2)x (r1 * r2) 0

x2 - (-6 10)x (-6 * 10) 0

x2 - 4x - 60 0

This confirms the derived equation:

x2 - 4x - 60 0

Conclusion and Verification

The quadratic equation for the given roots -6 and 10 is:

x2 - 4x - 60 0

This equation satisfies both the factors and the derived roots, validating the solution.