What is the Quadratic Equation for Roots -3 and -7?

When you are asked to find a quadratic equation with specific roots, such as -3 and -7, the problem can be solved using the relationships between the roots and the coefficients of the quadratic equation. This article will guide you through the process and provide a detailed explanation of the concept.

Understanding Quadratic Equations and Their Roots

A quadratic equation is a second-degree polynomial equation of the form:

ax2 bx c 0, where a, b, and c are constants, and a ≠ 0.

The roots of a quadratic equation are the values of ( x ) that satisfy the equation. If ( r_1 ) and ( r_2 ) are the roots of the quadratic equation, the equation can be expressed in the form:

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

Constructing a Quadratic Equation with Specific Roots

Given the roots ( r_1 -3 ) and ( r_2 -7 ), you can construct a quadratic equation by using the sum and product of the roots.

1. **Sum of the Roots**: ( r_1 r_2 -3 (-7) -10 )2. **Product of the Roots**: ( r_1 * r_2 (-3) * (-7) 21 )

Substitute these values into the general form of the quadratic equation:

x2 - (r1 r2)x r1 * r2 0

Resulting in:

x2 1 21 0

Verification and Variations

To verify, let’s check the roots of the equation ( x^2 1 21 0 ) using the quadratic formula ( x frac{-b pm sqrt{b^2 - 4ac}}{2a} ), where ( a 1 ), ( b 10 ), and ( c 21 ).

Solving this, we get:

x frac{-10 pm sqrt{10^2 - 4 * 1 * 21}}{2 * 1}

x frac{-10 pm sqrt{100 - 84}}{2}

x frac{-10 pm sqrt{16}}{2}

x frac{-10 pm 4}{2}

This gives us the roots ( x -3 ) and ( x -7 ), which confirms our solution.

It is important to note that there are infinitely many quadratic equations with the same roots, but they differ by a non-zero constant factor. For example:

1. ( 2x^2 2 42 0 )2. ( -x^2 - 1 - 21 0 )3. ( frac{x^2}{10} x frac{21}{10} 0 )

These equations are all equivalent in form, and they all have -3 and -7 as their roots.

Conclusion

The quadratic equation with roots -3 and -7 is ( x^2 1 21 0 ), and any equation of the form ( nx^2 10nx 21n 0 ), where ( n eq 0 ), is also a valid solution. Understanding the relationship between the roots and the coefficients is key to solving such problems.