Solving Quadratics

Quadratic equations are polynomial equations of degree 2, typically in the form $ax^{2} + bx + c = 0$. To solve these equations, we can use several methods, including factoring, completing the square, and the quadratic formula. The quadratic formula is given by:

$$x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$$

Example: Solve the quadratic equation $2x^{2} + 4x - 6 = 0$ using the quadratic formula.

Step 1: Identify coefficients:

• $a = 2$
• $b = 4$
• $c = -6$

Step 2: Calculate the discriminant $b^{2} - 4ac$:

$$4^{2} - 4(2)(-6) = 16 + 48 = 64$$

Step 3: Apply the quadratic formula:

$$x = \frac{-4 \pm \sqrt{64}}{2(2)}$$

Step 4: Simplify the square root:

$$x = \frac{-4 \pm 8}{4}$$

Step 5: Find the two possible solutions:

1. $x = \frac{-4 + 8}{4} = \frac{4}{4} = 1$
2. $x = \frac{-4 - 8}{4} = \frac{-12}{4} = -3$

Thus, the solutions to the equation $2x^{2} + 4x - 6 = 0$ are $x = 1$ and $x = -3$.

Key Questions:

• What is the standard form of a quadratic equation?
• How do you determine if a quadratic can be factored?
• What do the solutions of a quadratic equation represent?
• How does the discriminant affect the number of solutions?

Keywords:

• Quadratic equation
• Discriminant
• Quadratic formula
• Solutions
• Factoring