91Ó°ÊÓ

Factoring is a technique to solve quadratic equations when they can be expressed as a product of binomials. Given an equation in the form of \( ax^2 + bx + c = 0 \), factoring involves breaking it down into two binomials: \( (x - p)(x - q) = 0 \). These binomials represent solutions to the equation. To factor, find two numbers that multiply to \( c \) and add up to \( b \). For instance, in the equation \( x^2 - 7x + 12 = 0 \), we seek two numbers that multiply to 12 and add to -7. The numbers \(-3 \) and \(-4 \) work since \( -3 \times -4 = 12 \) and \( -3 + -4 = -7 \). Hence, we can write the equation as \( (x - 3)(x - 4) = 0 \) and then solve \( x = 3 \) or \( x = 4 \). Factoring is great for simple quadratic equations and provides a straightforward solution.

The quadratic formula is a universal method for solving any quadratic equation, whether it can be factored easily or not. For any quadratic equation \( ax^2 + bx + c = 0 \), the solutions can be found using the formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here’s how it works:

• Identify \( a \), \( b \), and \( c \) from the equation.
• Calculate the discriminant \( b^2 - 4ac \) to find the term under the square root.
• Substitute \( b \), the discriminant, \( a \), and \( c \) into the formula and solve.

Let's apply this to \( x^2 - 7x + 12 = 0 \). Here \( a = 1 \), \( b = -7 \), and \( c = 12 \). Plugging these in:

• Calculate the discriminant: \( (-7)^2 - 4(1)(12) = 49 - 48 = 1 \)
• Solve for \( x \): \( x = \frac{-(-7) \pm \sqrt{1}}{2(1)} = \frac{7 \pm 1}{2} \)
• Get two solutions \( x = 4 \) and \( x = 3 \).

The quadratic formula is highly reliable and often used when factoring is difficult.

Verifying solutions ensures that the values obtained solve the original equation correctly. It is an essential step to avoid mistakes. To verify a solution, substitute it back into the original quadratic equation and check if the expression equals zero. For the equation \( x^2 - 7x + 12 = 0 \), the solutions were \( x = 3 \) and \( x = 4 \). Let’s verify both:
- For \( x = 3 \):

• Substitute into the equation: \( 3^2 - 7(3) + 12 \)
• Simplify: \( 9 - 21 + 12 \)
• Result: \( 0 \)

- For \( x = 4 \):

• Substitute into the equation: \( 4^2 - 7(4) + 12 \)
• Simplify: \( 16 - 28 + 12 \)
• Result: \( 0 \)

Both substitutions simplify to zero, confirming that the solutions are correct. Always verify your solutions to ensure accuracy and understand why they work.