How to Factor Quadratic Equations: Simple Steps Explained
Quick Answer
To factor the quadratic equation $x^2 + 4x - 32 = 0$, find two numbers that multiply to -32 and add to 4. The numbers -4 and 8 work, leading to the factored form $(x - 4)(x + 8) = 0$.
Factoring quadratic equations is a fundamental skill in algebra that allows students to solve equations efficiently. Let's take a closer look at how to factor the quadratic equation $x^2 + 4x - 32 = 0$ step-by-step.
First, we start with the standard form of a quadratic equation, which is generally written as $ax^2 + bx + c = 0$. Here, our equation is $1x^2 + 4x - 32 = 0$, where $a = 1$, $b = 4$, and $c = -32$.
The goal of factoring is to express the quadratic equation as a product of two binomials. To do this, we need to find two numbers that meet two criteria: they must multiply to give the product of $a$ and $c$ (in this case, $1 imes -32 = -32$) and they must add up to $b$ (which is 4).
Next, we list the pairs of factors that multiply to -32:
- (1, -32)
- (-1, 32)
- (2, -16)
- (-2, 16)
- (4, -8)
- (-4, 8)
Now, we need to find a pair from this list that adds up to 4. Examining the pairs, we find that -4 and 8 satisfy our requirement since:
- -4 + 8 = 4
With these two numbers, we can rewrite the middle term (the 4x) of the equation. Thus, we can express the original quadratic equation as:
$$x^2 - 4x + 8x - 32 = 0$$
Now, we can group the terms:
$$(x^2 - 4x) + (8x - 32) = 0$$
Factoring each group gives:
$$x(x - 4) + 8(x - 4) = 0$$
Next, notice that both terms share a common factor of $(x - 4)$, which allows us to factor it out:
$$(x - 4)(x + 8) = 0$$
Now, we have successfully factored the quadratic equation. To find the solutions for $x$, we can set each factor equal to zero:
1. $x - 4 = 0$ → $x = 4$
2. $x + 8 = 0$ → $x = -8$
So, the solutions to the equation $x^2 + 4x - 32 = 0$ are $x = 4$ and $x = -8$.
Factoring quadratics is not only useful for solving equations but also plays a critical role in various real-world applications, such as physics and engineering, where quadratic relationships can model trajectories and other phenomena. Mastering this skill will greatly enhance your mathematical proficiency and problem-solving abilities.
For a more detailed breakdown of each step, you can refer to the equations section on our website.
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