Methods to Solve Quadratic Equations: Factoring and Quadratic Formula Explained

x^2 – 11x + 24

To work with the expression x^2 – 11x + 24, we can either factor it or find its roots using the quadratic formula

To work with the expression x^2 – 11x + 24, we can either factor it or find its roots using the quadratic formula. Let’s explore both methods:

Method 1: Factoring
To factor the quadratic expression x^2 – 11x + 24, we need to find two numbers that multiply to give 24 and add up to -11 (the coefficient of the x term).

We can list down all the pairs of factors of 24:
1, 24
2, 12
3, 8
4, 6

Since we need two numbers that add up to -11 (the coefficient of the x term), the pair of numbers that satisfies this condition is -3 and -8, as -3 + (-8) = -11.

Now, we can rewrite the quadratic expression by splitting the -11x term using -3x and -8x:
x^2 – 3x – 8x + 24

Then, we group the terms and factor them separately:
(x^2 – 3x) + (-8x + 24)
x(x – 3) – 8(x – 3)

Now, we can see that we have a common binomial factor of (x – 3) in both terms:
(x – 3)(x – 8)

Therefore, the factored form of x^2 – 11x + 24 is (x – 3)(x – 8).

Method 2: Quadratic Formula
If we want to find the roots of x^2 – 11x + 24 using the quadratic formula, we can use the general form of a quadratic equation:

x = (-b ± √(b^2 – 4ac)) / (2a)

In this case, the quadratic equation is x^2 – 11x + 24 = 0, so we have:
a = 1, b = -11, c = 24

Plugging these values into the quadratic formula:
x = (-(-11) ± √((-11)^2 – 4 * 1 * 24)) / (2 * 1)
x = (11 ± √(121 – 96)) / 2
x = (11 ± √25) / 2

Now, let’s consider both the positive and negative square root:
x = (11 + 5) / 2 or x = (11 – 5) / 2
x = 16 / 2 or x = 6 / 2
x = 8 or x = 3

Therefore, the roots of x^2 – 11x + 24 are x = 8 and x = 3.

Both factoring and using the quadratic formula can help us solve the given quadratic expression. You can choose the method that you feel comfortable with or that your teacher prefers.

More Answers: