Factoring the Quadratic Equation 2x^2 - 32x + 128 = 0
In this article, we will explore the process of factoring the quadratic equation 2x^2 - 32x + 128 = 0. Factoring is a powerful technique used to solve quadratic equations, and it can be a valuable tool in algebra and mathematics.
What is Factoring?
Factoring is the process of expressing a quadratic equation in the form:
ax^2 + bx + c = (dx + e)(fx + g)
where a, b, c, d, e, f, and g are constants. Factoring allows us to rewrite the quadratic equation as a product of two binomials, making it easier to solve.
Factoring the Equation 2x^2 - 32x + 128 = 0
To factor the equation 2x^2 - 32x + 128 = 0, we need to find two binomials that multiply to give us the original equation. Let's start by looking for two numbers whose product is 128 (the constant term) and whose sum is -32 (the coefficient of the x term).
These numbers are -8 and -16, since (-8)(-16) = 128 and (-8) + (-16) = -24, which is close to -32.
Now, we can write the factored form of the equation as:
2x^2 - 32x + 128 = (2x - 8)(x - 16) = 0
Solving the Factored Equation
To solve the equation, we set each factor equal to 0 and solve for x:
(2x - 8) = 0 --> 2x = 8 --> x = 4
(x - 16) = 0 --> x = 16
Therefore, the solutions to the equation 2x^2 - 32x + 128 = 0 are x = 4 and x = 16.
Conclusion
In this article, we have seen how to factor the quadratic equation 2x^2 - 32x + 128 = 0 using the technique of finding two binomials that multiply to give us the original equation. By factoring the equation, we were able to rewrite it as a product of two binomials, making it easier to solve. This technique can be applied to many other quadratic equations, and it is a powerful tool to have in your algebra toolkit.
