Step-by-Step Guide: Factoring and Using the Quadratic Formula to Solve x^2 – 11x + 24

x^2 – 11x + 24

To solve the expression x^2 – 11x + 24, we can factorize it or use the quadratic formula

To solve the expression x^2 – 11x + 24, we can factorize it or use the quadratic formula.

1. Factoring method:
To factorize the expression, we need to find two numbers whose product is 24 and whose sum is -11 (the coefficient of x).

The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.

Let’s try these factors:
1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24

Now, since the sum of the numbers should be -11, we can see that -3 and -8 will give us -11:
(-3) + (-8) = -11

Therefore, we can factorize the expression as:
x^2 – 11x + 24 = (x – 3)(x – 8)

2. Quadratic formula:
The quadratic formula is used to solve quadratic equations of the form ax^2 + bx + c = 0. In this case, a = 1, b = -11, and c = 24.

The quadratic formula states that the solutions for x are:
x = (-b ± sqrt(b^2 – 4ac)) / (2a)

Plugging in the values, we have:
x = (-(-11) ± sqrt((-11)^2 – 4(1)(24))) / (2(1))
x = (11 ± sqrt(121 – 96)) / 2
x = (11 ± sqrt(25)) / 2

Since sqrt(25) = 5, we further simplify:
x = (11 ± 5) / 2

This gives us two solutions for x:
x1 = (11 + 5) / 2 = 16 / 2 = 8
x2 = (11 – 5) / 2 = 6 / 2 = 3

So, the solutions to the equation x^2 – 11x + 24 = 0 are x = 3 and x = 8.

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