Question

Solve each equation. $$5(3 x-1)^{2}+3=-16(3 x-1)$$

   Solve each equation.
$$5(3 x-1)^{2}+3=-16(3 x-1)$$

Intermediate Algebra

Intermediate Algebra

Margaret L. Lial,… 11th Edition

Chapter 6, Problem 51

Instant Answer

Solved by Expert Heather Zimmers

07/25/2020

Step 1

Let's let $u$ equal $3x - 1$, the quantity in parentheses. This changes the equation to $5u^2 + 3 = -16u$. Show more…

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Solve each equation. $$5(3 x-1)^{2}+3=-16(3 x-1)$$

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Key Concepts

Substitution Method

The substitution method involves replacing a complex expression within an equation with a simpler variable, which simplifies the process of solving the equation. It is particularly useful when the same expression appears in multiple parts of an equation, effectively reducing the original problem into a more manageable form.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation typically expressed in the standard form ax² + bx + c = 0. Understanding the nature of quadratic equations is fundamental in algebra, as they often appear in various mathematical contexts and can be solved by several methods.

Standard Form of a Quadratic Equation

Expressing a quadratic equation in standard form is essential because it sets up the equation for solution methods such as factoring, completing the square, or applying the quadratic formula. It standardizes the structure, making it easier to identify the coefficients and apply the appropriate solving techniques.

Methods for Solving Quadratic Equations

Once a quadratic equation is in standard form, it can be solved using methods such as factoring when the quadratic is factorable, or by applying the quadratic formula. These methods provide a systematic way to find the roots of the equation and are a key component of algebraic problem-solving.

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