A quadratic equation is a type of polynomial equation that has the form:

ax² + bx + c = 0

In this equation:

• a is the coefficient of x² (the quadratic term).
• b is the coefficient of x (the linear term).
• c is the constant (the term without an x).

What Makes an Equation Quadratic?

The defining characteristic of a quadratic equation is that the highest power of the variable is 2 (x²). This means that the equation will have a curve when graphed, which is called a parabola. Quadratic equations can have 0, 1, or 2 solutions, depending on the discriminant (b² - 4ac) in the quadratic formula.

How to Solve a Quadratic Equation?

There are a few methods to solve quadratic equations:

• Factoring: Express the quadratic as a product of two binomials.
• Quadratic Formula: Use the formula x = (-b ± √(b² - 4ac)) / 2a to find the solutions.
• Completing the Square: Rewrite the equation in the form (x + p)² = q and solve for x.

Example 1: Solving by Factoring

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

Step 1: 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.
Step 2: Factor the quadratic equation: (x - 2)(x - 3) = 0
Step 3: Set each factor equal to zero and solve for x:

• x - 2 = 0 ⟹ x = 2
• x - 3 = 0 ⟹ x = 3

So, the solutions are x = 2 and x = 3.

Example 2: Solving using the Quadratic Formula

Solve the quadratic equation: 2x² + 4x - 6 = 0
Step 1: Identify a, b, and c from the equation: a = 2, b = 4, c = -6
Step 2: Apply the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Step 3: Substitute the values into the formula:

x = (-(4) ± √(4² - 4(2)(-6))) / 2(2)

x = (-4 ± √(16 + 48)) / 4

x = (-4 ± √64) / 4

x = (-4 ± 8) / 4

Step 4: Solve for the two possible values of x:

• x = (-4 + 8) / 4 = 4 / 4 = 1
• x = (-4 - 8) / 4 = -12 / 4 = -3

So, the solutions are x = 1 and x = -3.

Summary:

A quadratic equation is any equation in the form ax² + bx + c = 0. To solve these equations, you can use factoring, the quadratic formula, or completing the square. Quadratic equations can have 0, 1, or 2 real solutions depending on the discriminant.

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