What Is A Root In A Quadratic Equation

Understanding roots in quadratic equations is fundamental for anyone delving into algebra. These roots, also known as solutions or zeros, are the values of the variable that satisfy the equation, making the expression equal to zero. Let’s explore the concept of roots in quadratic equations in detail.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:

[ ax^2 + bx + c = 0 ]

where:

• ( x ) represents the variable or unknown.
• ( a ), ( b ), and ( c ) are constants, with ( a \neq 0 ).
• ( a ) is the quadratic coefficient.
• ( b ) is the linear coefficient.
• ( c ) is the constant term.

The "roots" of a quadratic equation are the values of ( x ) that make the equation true, i.e., the values of ( x ) that satisfy ( ax^2 + bx + c = 0 ). These roots can be real or complex numbers.

Methods to Find Roots of Quadratic Equations

There are several methods to find the roots of a quadratic equation, each with its own advantages and applications:

1. Factoring
2. Completing the Square
3. Quadratic Formula

Let's explore each method in detail.

1. Factoring

Factoring involves expressing the quadratic equation as a product of two binomials. If the quadratic equation can be factored, this method is often the quickest way to find the roots.

Steps for Factoring:

1. Rewrite the equation: Ensure the quadratic equation is in the standard form ( ax^2 + bx + c = 0 ).
2. Factor the quadratic expression: Find two numbers that multiply to ( ac ) (the product of ( a ) and ( c )) and add up to ( b ).
3. Rewrite the middle term: Replace ( bx ) with the sum of the two numbers found in the previous step.
4. Factor by grouping: Group the terms and factor out the common factors.
5. Set each factor equal to zero: Solve for ( x ) to find the roots.

Example: Consider the quadratic equation: [ x^2 - 5x + 6 = 0 ]

1. The equation is already in the standard form.
2. We need to find two numbers that multiply to ( 6 ) (the constant term) and add up to ( -5 ) (the coefficient of ( x )). These numbers are ( -2 ) and ( -3 ).
3. Rewrite the middle term: [ x^2 - 2x - 3x + 6 = 0 ]
4. Factor by grouping: [ x(x - 2) - 3(x - 2) = 0 ] [ (x - 2)(x - 3) = 0 ]
5. Set each factor equal to zero: [ x - 2 = 0 \quad \text{or} \quad x - 3 = 0 ] [ x = 2 \quad \text{or} \quad x = 3 ]

Thus, the roots of the quadratic equation ( x^2 - 5x + 6 = 0 ) are ( x = 2 ) and ( x = 3 ).

2. Completing the Square

Completing the square is a method used to convert a quadratic equation into a perfect square trinomial, which can then be easily solved. This method is particularly useful when the quadratic equation cannot be easily factored.

Steps for Completing the Square:

1. Rewrite the equation: Ensure the quadratic equation is in the standard form ( ax^2 + bx + c = 0 ).
2. Divide by ( a ): If ( a \neq 1 ), divide the entire equation by ( a ) to make the coefficient of ( x^2 ) equal to 1.
3. Move the constant term to the other side: Rewrite the equation in the form ( x^2 + \frac{b}{a}x = -\frac{c}{a} ).
4. Add the square of half the coefficient of ( x ) to both sides: Add ( \left(\frac{b}{2a}\right)^2 ) to both sides of the equation.
5. Factor the perfect square trinomial: The left side of the equation is now a perfect square trinomial and can be factored as ( \left(x + \frac{b}{2a}\right)^2 ).
6. Take the square root of both sides: Solve for ( x ) by taking the square root of both sides and isolating ( x ).

Example: Consider the quadratic equation: [ x^2 - 6x + 5 = 0 ]

1. The equation is already in the standard form.
2. The coefficient of ( x^2 ) is already 1.
3. Move the constant term to the other side: [ x^2 - 6x = -5 ]
4. Add the square of half the coefficient of ( x ) to both sides: [ \left(\frac{-6}{2}\right)^2 = (-3)^2 = 9 ] [ x^2 - 6x + 9 = -5 + 9 ]
5. Factor the perfect square trinomial: [ (x - 3)^2 = 4 ]
6. Take the square root of both sides: [ x - 3 = \pm \sqrt{4} ] [ x - 3 = \pm 2 ]
7. Solve for ( x ): [ x = 3 \pm 2 ] [ x = 3 + 2 \quad \text{or} \quad x = 3 - 2 ] [ x = 5 \quad \text{or} \quad x = 1 ]

Thus, the roots of the quadratic equation ( x^2 - 6x + 5 = 0 ) are ( x = 5 ) and ( x = 1 ).

3. Quadratic Formula

The quadratic formula is a universal method for finding the roots of any quadratic equation. It is derived from the method of completing the square and can be applied to any quadratic equation in the form ( ax^2 + bx + c = 0 ).

Quadratic Formula: The roots of the quadratic equation ( ax^2 + bx + c = 0 ) are given by: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

Steps for Using the Quadratic Formula:

1. Identify ( a ), ( b ), and ( c ): Determine the values of ( a ), ( b ), and ( c ) from the quadratic equation ( ax^2 + bx + c = 0 ).
2. Substitute into the formula: Plug the values of ( a ), ( b ), and ( c ) into the quadratic formula.
3. Simplify: Simplify the expression to find the roots.

Example: Consider the quadratic equation: [ 2x^2 + 3x - 5 = 0 ]

1. Identify ( a ), ( b ), and ( c ): [ a = 2, \quad b = 3, \quad c = -5 ]
2. Substitute into the formula: [ x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-5)}}{2(2)} ]
3. Simplify: [ x = \frac{-3 \pm \sqrt{9 + 40}}{4} ] [ x = \frac{-3 \pm \sqrt{49}}{4} ] [ x = \frac{-3 \pm 7}{4} ]
4. Solve for ( x ): [ x = \frac{-3 + 7}{4} \quad \text{or} \quad x = \frac{-3 - 7}{4} ] [ x = \frac{4}{4} \quad \text{or} \quad x = \frac{-10}{4} ] [ x = 1 \quad \text{or} \quad x = -\frac{5}{2} ]

Thus, the roots of the quadratic equation ( 2x^2 + 3x - 5 = 0 ) are ( x = 1 ) and ( x = -\frac{5}{2} ).

The Discriminant

The discriminant is the part of the quadratic formula under the square root sign, i.e., ( b^2 - 4ac ). The discriminant provides valuable information about the nature of the roots of the quadratic equation.

Discriminant: [ \Delta = b^2 - 4ac ]

The discriminant can be used to determine whether the roots are:

• Real and distinct: If ( \Delta > 0 ), the quadratic equation has two distinct real roots.
• Real and equal: If ( \Delta = 0 ), the quadratic equation has two equal real roots (or one repeated real root).
• Complex: If ( \Delta < 0 ), the quadratic equation has two complex conjugate roots.

Examples:

1. Real and distinct roots: Consider the quadratic equation ( x^2 - 5x + 6 = 0 ). [ a = 1, \quad b = -5, \quad c = 6 ] [ \Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1 ] Since ( \Delta > 0 ), the equation has two distinct real roots.

2. Real and equal roots: Consider the quadratic equation ( x^2 - 4x + 4 = 0 ). [ a = 1, \quad b = -4, \quad c = 4 ] [ \Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0 ] Since ( \Delta = 0 ), the equation has two equal real roots.

3. Complex roots: Consider the quadratic equation ( x^2 + 2x + 5 = 0 ). [ a = 1, \quad b = 2, \quad c = 5 ] [ \Delta = (2)^2 - 4(1)(5) = 4 - 20 = -16 ] Since ( \Delta < 0 ), the equation has two complex roots.

Properties of Roots

The roots of a quadratic equation have several interesting properties. Let ( x_1 ) and ( x_2 ) be the roots of the quadratic equation ( ax^2 + bx + c = 0 ). Then:

1. Sum of the roots: [ x_1 + x_2 = -\frac{b}{a} ]
2. Product of the roots: [ x_1 \cdot x_2 = \frac{c}{a} ]

These properties can be useful for:

• Verifying the roots found.
• Constructing a quadratic equation given its roots.
• Solving problems involving the roots without actually finding them.

Example: Consider the quadratic equation ( x^2 - 5x + 6 = 0 ). The roots are ( x_1 = 2 ) and ( x_2 = 3 ).

1. Sum of the roots: [ x_1 + x_2 = 2 + 3 = 5 ] [ -\frac{b}{a} = -\frac{-5}{1} = 5 ]
2. Product of the roots: [ x_1 \cdot x_2 = 2 \cdot 3 = 6 ] [ \frac{c}{a} = \frac{6}{1} = 6 ]

Thus, the properties of the roots are verified.

Complex Roots

When the discriminant ( \Delta = b^2 - 4ac ) is negative, the quadratic equation has complex roots. Complex roots always occur in conjugate pairs, i.e., if ( \alpha + i\beta ) is a root, then ( \alpha - i\beta ) is also a root, where ( \alpha ) and ( \beta ) are real numbers, and ( i ) is the imaginary unit, defined as ( i = \sqrt{-1} ).

Example: Consider the quadratic equation ( x^2 + 2x + 5 = 0 ). [ a = 1, \quad b = 2, \quad c = 5 ] [ \Delta = (2)^2 - 4(1)(5) = 4 - 20 = -16 ] Since ( \Delta < 0 ), the equation has complex roots. Using the quadratic formula: [ x = \frac{-2 \pm \sqrt{-16}}{2(1)} ] [ x = \frac{-2 \pm 4i}{2} ] [ x = -1 \pm 2i ] The roots are ( x_1 = -1 + 2i ) and ( x_2 = -1 - 2i ), which are complex conjugates.

Applications of Roots of Quadratic Equations

Understanding and finding the roots of quadratic equations has numerous applications in various fields, including:

1. Physics: In projectile motion, quadratic equations are used to model the trajectory of an object, and the roots of the equation give the time at which the object reaches a certain height or the range of the projectile.
2. Engineering: In structural engineering, quadratic equations are used to calculate stresses and strains in materials.
3. Economics: In finance, quadratic equations can be used to model cost functions, revenue functions, and profit maximization problems.
4. Computer Science: In computer graphics and game development, quadratic equations are used to model curves and surfaces.

Graphical Interpretation of Roots

Graphically, the roots of a quadratic equation ( ax^2 + bx + c = 0 ) represent the ( x )-intercepts of the parabola ( y = ax^2 + bx + c ).

• Two distinct real roots: The parabola intersects the ( x )-axis at two distinct points.
• One real root (repeated): The parabola touches the ( x )-axis at one point (the vertex of the parabola lies on the ( x )-axis).
• No real roots (complex roots): The parabola does not intersect the ( x )-axis.

Understanding the graphical interpretation of roots can provide valuable insights into the behavior of quadratic functions.

Conclusion

The roots of a quadratic equation are fundamental to understanding quadratic functions and their applications. By mastering the methods to find these roots—factoring, completing the square, and using the quadratic formula—one can solve a wide range of problems in mathematics, science, and engineering. The discriminant provides valuable information about the nature of the roots, whether they are real and distinct, real and equal, or complex. Understanding the properties of roots, such as the sum and product of roots, and the graphical interpretation of roots further enhances one's understanding of quadratic equations and their significance in various fields.