Problem 63 - Introductory Algebra for College Students

Question

Solve each equation and check your solutions. $$2(x-4)^{2}+x^{2}=x(x+50)-46 x$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 2\) and \(x = 8\).

Step 1: Expand and Simplify

Expand the term \(2(x-4)^{2}\) which yields \(2(x^{2} - 8x + 16)\) and simplifies to \(2x^{2} - 16x + 32\). Next, distribute the terms \(x(x+50)\) and \(- 46x\) which yields \(x^{2} + 50x -46x\). The original equation now transforms into:\(2x^{2} - 16x + 32 + x^{2} = x^{2} + 50x - 46x\).

Step 2: Collect and Simplify Like Terms

Combine like terms to simplify the equation into a standard quadratic form. The equation becomes \(3x^{2} - 16x + 32 = x^{2} + 4x\). Move all terms to one side and simplify to get \(2x^{2} - 20x + 32 = 0\) which is now a quadratic equation in standard form.

Step 3: Solve for x using Quadratic Formula

The quadratic formula is \((-b ± sqrt{b^{2} - 4ac}) / (2a)\). From the equation \(2x^{2} - 20x + 32 = 0\), \(a = 2, b = -20, c = 32\). Substitute a, b and c into the quadratic formula, and solve for \(x\), which yields \(x = 2, x = 8\).

Step 4: Check Solutions

Substitute both \(x = 2\) and \(x = 8\) back into the original equation separately to ensure that both solutions satisfy the equation so that the left hand side equals the right side.

Key Concepts

Quadratic Formula

The quadratic formula is a crucial tool in solving quadratic equations of the form \(ax^2 + bx + c = 0\). This formula allows us to find the values of \(x\) without needing to factor the quadratic, which can sometimes be complex or even impossible to do easily by inspection. The formula is given as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where:

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
• \(b^2 - 4ac\) is known as the discriminant. It determines the nature of the roots of the equation (real and distinct, real and repeated, or complex).
• The symbol \(\pm\) indicates that there are usually two solutions.

To solve the equation using the quadratic formula, simply substitute the coefficients \(a\), \(b\), and \(c\) from your given equation into the formula, calculate the discriminant, and then find the values of \(x\). Checking if your solutions are correct by substituting them back into the original equation is a wise step.

Expanding Expressions

Expanding expressions involves using algebraic methods to transform an expression like \((x - 4)^2\) into a more extended form. This process is essential in simplifying and solving equations. For expansion, you multiply each term inside the parentheses by itself or by an external multiplier.For example, consider \(2(x - 4)^2\). First, expand \((x - 4)^2\) which is:

• \((x - 4)(x - 4) = x^2 - 4x - 4x + 16 = x^2 - 8x + 16\)

Once expanded, multiply the entire expression by \(2\), resulting in \(2x^2 - 16x + 32\). Expanding allows us to re-frame equations from one form to another, making subsequent steps like solving using the quadratic formula much more manageable.

Simplifying Equations

Simplifying equations is all about reducing them to their simplest form, which often makes them easier to solve. This can involve combining like terms, distributing factors, and canceling terms where possible. It's an integral part of ensuring your math problems are as straightforward as possible to handle further.In our exercise, by moving all terms to one side, the equation \(3x^2 - 16x + 32 = x^2 + 4x\) is simplified by collecting like terms:

• Combine like terms \(3x^2\) and \(x^2\) to get \(2x^2\).
• Combine \(-16x\) and \(4x\) to get \(-20x\).

This simplification leads to a more standard form suitable for applying the quadratic formula: \(2x^2 - 20x + 32 = 0\). Simplification helps in reducing clutter, making computations more direct and revealing the underlying structure of the equations.