Solving Quadratic Equations One Variable

 Definition of a Quadratic Equation

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

ax² + bx + c = 0

where:

a,b, and care constants (with  a ≠ 0),

x represents the variable.

 

Quadratic equations are fundamental in algebra and appear in various real-life applications, such as physics, engineering, and finance.

 

How to Solve Quadratic Equations

There are several methods to solve quadratic equations, including:

1. Factoring Method

If a quadratic equation can be factored into two binomials, we can set each factor equal to zero to find the values of x.

 

2. Quadratic Formula

The quadratic formula provides a direct way to find the roots of any quadratic equation:

The term under the square root, b²−4ac is called the discriminant.

It determines the nature of the roots:

• If b² − 4ac > 0, there are two distinct real roots.
• If b² − 4ac = 0, there is one real root (a repeated root).
• If b² − 4ac < 0, there are two complex (imaginary) roots.

 

3. Completing the Square

This method involves rewriting the quadratic equation in the form (x − p)² = q and then solving for x by taking the square root on both sides.

 

4. Graphical Method

By plotting the quadratic equation as y = ax²+ bx +c, the solutions correspond to the points where the graph intersects the x-axis.

 

Example Problems

Example 1: Solving by Factoring

Solve x² − 5x + 6 = 0.

Solution:

1. Factor the quadratic equation: (x + p)(x + q) = 0

            p + q = -5 and pq = 6

            Obtaine the value p = -2 and q = -3

            Then (x − 2)(x − 3) = 0

2. Set each factor to zero: x − 2 = 0 or x − 3 = 0.

3. Solve for x:  x = 2 or x = 3 

Thus, the solutions are x₁ = 2 or x₂ = 3.

 

Example 2: Solving by Factoring

Solve x² + 5x − 14 = 0.

Solution:

1. Factor the quadratic equation: (x + p)(x + q) = 0

            p + q = 5 and pq = -14

            Obtaine the value p = -2 and q = 7

            Then (x − 2)(x + 7) = 0

2. Set each factor to zero: x − 2 = 0 or x + 7 = 0.

3. Solve for x:  x = 2 or x = -7 

Thus, the solutions are x₁ = 2 or x₂ = -7.

Example 3: Solving Using the Quadratic Formula

Solve x² − 6x + 2 = 0 using the quadratic formula.

Solution:

1. Identify coefficients: a = 1, b = −6, c = 2.

2. Compute the discriminant: b² − 4ac = (−6)² − 4(1)(2) = 36 - 8 = 28

3. Apply the quadratic formula:

Example 4: Solving Using the Quadratic Formula

Solve 2x² − 4x − 3 = 0 using the quadratic formula.

Solution:

1. Identify coefficients: a = 2, b = −4, c = -5.

2. Compute the discriminant: b² − 4ac = (−4)² − 4(2)(−3) = 16 + 24 = 40

3. Apply the quadratic formula:

Conclusion

Quadratic equations can be solved using multiple methods depending on the situation. Factoring is efficient when possible, the quadratic formula is reliable for all cases, and completing the square provides insight into the equation's structure. Mastering these techniques is essential for success in algebra and beyond.