Solving simple quadratic equations involves finding the values of $x$ that satisfy the equation. A quadratic equation has the standard form:

$ax^2 + bx + c = 0$

where:

• $a, b,$ and $c$ are constants,
• $a \neq 0$,
• $x$ is the variable.

Methods to Solve Simple Quadratic Equations

1. Factoring (When the quadratic can be factored easily)
2. Quadratic Formula (Always works)
3. Completing the Square (Useful for understanding transformations)
4. Square Root Method (For equations without a linear term)

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1. Solving by Factoring

This method works when the quadratic expression can be factored into two binomials.

Example: $x^2 - 5x + 6 = 0$ Factor the quadratic expression: $(x - 2)(x - 3) = 0$ Set each factor to zero: $x - 2 = 0 \quad \text{or} \quad x - 3 = 0$ $x = 2, \quad x = 3$

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2. Solving Using the Quadratic Formula

If factoring is difficult, use the quadratic formula:

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

Example: $2x^2 - 4x - 6 = 0$

Identify coefficients: $a = 2, \quad b = -4, \quad c = -6$

Calculate the discriminant: $b^2 - 4ac = (-4)^2 - 4(2)(-6) = 16 + 48 = 64$

Apply the quadratic formula: $x = \frac{4 \pm \sqrt{64}}{4}$

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

$x = 3 \quad \text{or} \quad x = -1$

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3. Solving by Completing the Square

This method is useful when the quadratic equation doesn't factor easily.

Example: $x^2 + 6x + 5 = 0$

Move the constant to the other side: $x^2 + 6x = -5$

Take half of the coefficient of $x$, square it, and add to both sides: $\left(\frac{6}{2}\right)^2 = 9$

$x^2 + 6x + 9 = -5 + 9$

Rewrite as a square: $(x + 3)^2 = 4$

Solve by taking the square root: $x + 3 = \pm 2$

$x = -3 \pm 2$

$x = -1, \quad x = -5$

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4. Solving Using the Square Root Method

Use this method when there is no linear term ($bx = 0$).

Example: $x^2 = 9$

Take the square root of both sides: $x = \pm\sqrt{9}$

$x = \pm 3$

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Do you need a detailed explanation on any method?

Here are some related questions for deeper understanding:

1. How do you determine which method is best for solving a quadratic equation?
2. What does the discriminant tell us about the number of solutions?
3. How does completing the square help in deriving the quadratic formula?
4. Can all quadratic equations be factored? Why or why not?
5. How are quadratic equations used in real-world applications?

Tip: Always check your solutions by substituting them back into the original equation!