Solving quadratic equations

Solving Quadratic Equations

A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to this equation are known as the roots of the equation. There are several methods to solve quadratic equations: factoring, using the quadratic formula, and completing the square. This lesson will cover each method in detail.

Factoring Quadratic Equations

Factoring involves rewriting the quadratic equation in a product form. If the equation can be factored, it will be expressed as (px + q)(rx + s) = 0. The solutions can then be found by setting each factor equal to zero.

Consider the quadratic equation:

2x² + 8x = 0

To solve this equation by factoring, first, factor out the common term:

2x(x + 4) = 0

Now, set each factor equal to zero:

2x = 0   or   x + 4 = 0

From the first factor:

x = 0

From the second factor:

x + 4 = 0  =>  x = -4

The solutions to the equation are:

x = 0   and   x = -4

The Quadratic Formula

The quadratic formula is a general method for solving any quadratic equation. It is given by:

x = (-b ± √(b² - 4ac)) / (2a)

Here, b² - 4ac is called the discriminant. The discriminant 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 no real roots (the roots are complex).

Let’s solve a quadratic equation using the quadratic formula:

Consider:

x² - 6x + 8 = 0

Here, a = 1, b = -6, and c = 8. First, calculate the discriminant:

Discriminant = b² - 4ac = (-6)² - 4(1)(8) = 36 - 32 = 4

Since the discriminant is greater than zero, we have two distinct real roots. Now, apply the quadratic formula:

x = (6 ± √4) / (2 × 1)

Calculate the roots:

x = (6 ± 2) / 2

This gives us:

x = (6 + 2) / 2 = 8 / 2 = 4

x = (6 - 2) / 2 = 4 / 2 = 2

The solutions to the equation are:

x = 4   and   x = 2

Completing the Square

Completing the square is another method to solve quadratic equations. This method involves rewriting the equation in the form (x - p)² = q. Here is how to do it:

Consider the equation:

x² + 4x - 5 = 0

First, move the constant term to the other side:

x² + 4x = 5

Next, complete the square. Take half of the coefficient of x (which is 4), square it, and add it to both sides:

(4/2)² = 2² = 4

Add 4 to both sides:

x² + 4x + 4 = 5 + 4

This simplifies to:

(x + 2)² = 9

Now, take the square root of both sides:

x + 2 = ±√9

This gives:

x + 2 = 3   or   x + 2 = -3

Now, solve for x:

x = 3 - 2 = 1   or   x = -3 - 2 = -5

The solutions to the equation are:

x = 1   and   x = -5

Summary

• A quadratic equation is of the form ax² + bx + c = 0.
• Methods to solve quadratic equations include factoring, the quadratic formula, and completing the square.
• The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a).
• The discriminant b² - 4ac determines the nature of the roots.

Check your understanding

1. What is the quadratic formula?
2. How can you determine the number of real roots from the discriminant?
3. Factor the equation x² - 5x + 6 = 0.
4. Use the completing the square method to solve x² + 6x + 8 = 0.