Quadratic equation

Quadratic Equations:

Quadratic equations are algebraic equations of the form:

ax² + bx + c = 0

Here, x represents the variable, and a, b, and c are constants, with a ≠ 0. The highest power of the variable in a quadratic equation is 2, making it a second-degree polynomial equation.

The solutions to a quadratic equation are the values of x that satisfy the equation and make it true. Quadratic equations can have zero, one, or two real solutions, or two complex solutions

Quadratic Formula:

1. Solving Quadratic Equations by Factorization:

The quadratic formula is a formula that provides the solutions to any quadratic equation of the form

ax² + bx + c = 0.

It is derived from completing the square and is given by:

\(x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)

In this formula:

x represents the variable or unknown values that satisfy the quadratic equation.
a, b, and c are constants, with a ≠ 0, that represent the coefficients of the quadratic equation.
± denotes that there are two possible solutions: one with a plus sign and one with a minus sign.

In this method, we try to express the quadratic equation as a product of two binomials and then set each binomial equal to zero to find the values of the variable.

Step-by-Step Process:

Write the quadratic equation in standard form: ax² + bx + c = 0, where a, b, and c are constants.Look for two binomials whose product equals the quadratic equation.Set each binomial equal to zero and solve for the variable

Example :

Solve the quadratic equation: x² – 5x + 6 = 0.

Solution:

Step 1: The quadratic equation is already in standard form.

Step 2: Look for two binomials whose product equals x² – 5x + 6. We have (x – 2)(x – 3).

Step 3: Set each binomial equal to zero and solve for x:

x – 2 = 0 –> x = 2

x – 3 = 0 –> x = 3

Use the distributive property: 2(a + 3) = 2a + 6, and -4(2a – 1) = -8a + 4.

Combine like terms: 2a – 8a = -6a.

The simplified expression is -6a + 6.

2. Solving Quadratic Equations by Completing the Square:

In this method, we transform the quadratic equation into a perfect square trinomial by adding or subtracting a constant and then solving for the variable.

Step-by-Step Process:

Write the quadratic equation in standard form: ax² + bx + c = 0, where a, b, and c are constants.

Complete the square by adding or subtracting a constant to both sides of the equation to form a perfect square trinomial on the left side.

Factor the perfect square trinomial and solve for the variable.

Examples :

Solve the quadratic equation: x² + 6x + 8 = 0.

Solution:

Step 1: The quadratic equation is already in standard form.

Step 2: Complete the square by adding 4 to both sides of the equation: x² + 6x + 4 + 4 = 4.

Step 3: Factor the perfect square trinomial and solve for x:

(x + 2)(x + 2) = 4

(x + 2)² = 4

x + 2 = ±√4

x + 2 = ±2

x = -2 ± 2

The solutions are x = 0 and x = -4.

Both factorization and completing the square methods are powerful techniques to solve quadratic equations. However, completing the square method can be more useful when the quadratic equation does not factor easily. It’s essential to practice both methods to become proficient in solving quadratic equations and apply them to various mathematical problems.

Solving Quadratic Equations by Factorization & Completing Square Method

Quadratic equation