Exploring Quadratic Equations with Roots 7 and -3

Quadratic equations are a fundamental concept in algebra, often appearing in various fields such as physics, engineering, and economics. A quadratic equation is an equation of the form ( ax^2 bx c 0 ), where ( a, b, ) and ( c ) are constants, and ( a eq 0 ). In this detailed exploration, we delve into a specific type of quadratic equation: one that has roots 7 and -3.

Understanding Quadratic Equations

Before we dive into the specifics, let's review what a quadratic equation is. A quadratic equation is an equation that can be written in the form:

[ ax^2 bx c 0 ]

where ( a, b, ) and ( c ) are constants, and ( a eq 0 ). The term ( ax^2 ) is the quadratic term, ( bx ) is the linear term, and ( c ) is the constant term. The roots of a quadratic equation are the values of ( x ) that satisfy the equation.

Quadratic Equation with Roots 7 and -3

Given the roots of a quadratic equation, we can construct the equation using the fact that if ( r_1 ) and ( r_2 ) are the roots of the equation, then the quadratic equation can be written as:

[ (x - r_1)(x - r_2) 0 ]

For the roots 7 and -3, we can set up the equation as follows:

[ (x - 7)(x 3) 0 ]

Expanding this expression gives:

[ x^2 3x - 7x - 21 0 ]

[ x^2 - 4x - 21 0 ]

Therefore, the quadratic equation with roots 7 and -3 is ( x^2 - 4x - 21 0 ).

Solving the Quadratic Equation

To solve the quadratic equation ( x^2 - 4x - 21 0 ), we can use the quadratic formula:

[ x frac{-b pm sqrt{b^2 - 4ac}}{2a} ]

For our equation, ( a 1 ), ( b -4 ), and ( c -21 ). Plugging these values into the quadratic formula gives:

[ x frac{-(-4) pm sqrt{(-4)^2 - 4(1)(-21)}}{2(1)} ]

[ x frac{4 pm sqrt{16 84}}{2} ]

[ x frac{4 pm sqrt{100}}{2} ]

[ x frac{4 pm 10}{2} ]

Which yields two solutions:

[ x_1 frac{4 10}{2} 7 ]

[ x_2 frac{4 - 10}{2} -3 ]

Thus, the roots of the quadratic equation ( x^2 - 4x - 21 0 ) are indeed 7 and -3, as expected.

Applications of Quadratic Equations

Quadratic equations with roots 7 and -3, like many other quadratic equations, have wide-ranging applications. For example:

Physics: They can model the motion of objects under the influence of gravity, such as the trajectory of a projectile. Engineering: They can be used in designing structures and systems, such as calculating the strength of beams or cables. Economics: They can represent cost and revenue functions, helping to find the break-even point or maximize profits.

Summary

In this article, we have explored the quadratic equation with roots 7 and -3. We have seen how to construct such an equation from its roots and how to solve the equation using the quadratic formula. Additionally, we have discussed the practical applications of quadratic equations in various fields.

Understanding quadratic equations is crucial for students and professionals alike, as it opens up a world of problem-solving possibilities in numerous disciplines. Whether you are a high school student or a professional in a related field, mastering quadratic equations can significantly enhance your problem-solving skills.