Problem 74 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$(x+3) 2-10(x+5)=-2(x+1)$$

Step-by-Step Solution

Answer

Solutions are \(x = 1 + 2\sqrt{10}\) and \(x = 1 - 2\sqrt{10}\).

Step 1: Expand and Simplify the Equation

Firstly, expand the terms of the equation. Expand \((x+3)^2\) which becomes x^2 + 6x + 9.Then, expand \(-10(x+5)\) which becomes -10x - 50.The equation becomes:\[x^2 + 6x + 9 - 10x - 50 = -2(x+1)\]

Step 2: Reorganize the Equation

Distribute on the right-hand side as well:\[-2(x+1)\] becomes -2x - 2.Then combine like terms from both sides to get:\[x^2 - 4x - 41 = -2x - 2\]

Step 3: Rearrange the Equation into Standard Form

Bring all terms to one side to prepare for using the quadratic formula:\[x^2 - 4x - 41 + 2x + 2 = 0\]This simplifies to:\[x^2 - 2x - 39 = 0\]

Step 4: Identify A, B, and C

Identify the coefficients in the standard quadratic form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -2\), \(c = -39\).

Step 5: Apply the Quadratic Formula

Use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute \(a = 1\), \(b = -2\), \(c = -39\), resulting in:\[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-39)}}{2(1)}\]Simplify inside the square root:\[x = \frac{2 \pm \sqrt{4 + 156}}{2}\]\[x = \frac{2 \pm \sqrt{160}}{2}\]

Step 6: Simplify the Roots

Now, simplify \(\sqrt{160}\) which is \(4\sqrt{10}\). Thus, the roots simplify to:\[x = \frac{2 \pm 4\sqrt{10}}{2}\]This simplifies further to:\[x = 1 \pm 2\sqrt{10}\]

Step 7: State the Solution

The solutions to the quadratic equation are \(x = 1 + 2\sqrt{10}\) and \(x = 1 - 2\sqrt{10}\).

Key Concepts

Quadratic Equation

A quadratic equation is any equation that can be rearranged in the form \( ax^2 + bx + c = 0 \). In this form, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \) to ensure it remains a quadratic equation. What makes these equations special is the \( x^2 \) term, representing a parabolic shape when graphed. It extends infinitely and can open upwards or downwards depending on the sign of \( a \). Quadratic equations are pivotal in algebra as they connect linear and higher degree polynomial equations. Applications include projectile motion, area optimization, and physics problems where variables change quadratically with time or space. Quadratic equations can be solved using different methods such as factoring, graphing, completing the square, and the quadratic formula.

Completing the Square

Completing the square is a mathematical technique used to simplify solving quadratic equations. It rearranges a standard quadratic equation such that it becomes a perfect square trinomial. This method helps derive the quadratic formula and understand the equation's properties, such as its vertex form for graphing. To complete the square:

• Start with a quadratic equation \( ax^2 + bx + c = 0 \), and divide everything by \( a \) if \( a eq 1 \).
• Move \( c \) to the opposite side of the equation.
• Add \( (\frac{b}{2})^2 \) to both sides to balance the equation, making the left side a perfect square trinomial.
• This will transform it into \((x + \frac{b}{2})^2 = d \), where \( d \) is the adjusted constant on the right.
• Finally, solve for \( x \) by taking the square root of both sides and solving the resulting linear equation.

Through completing the square, students gain deeper insights into the nature of quadratics and the symmetry of parabolas.

Solving Equations

Solving equations in algebra involves finding the values of unknown variables that satisfy the equation. In the context of quadratic equations, this means determining which values of \( x \) make the equation true. Quadratic equations typically have two solutions due to their nature. In solving them, one must:

• Start by transforming the quadratic equation into the form \( ax^2 + bx + c = 0 \).
• Use appropriate methods like factoring, completing the square, or the quadratic formula depending on the specific equation and its coefficients.
• Verify solutions by substituting them back into the original equation to ensure they satisfy it.

The aim is not just to compute results, but to understand underlying problem conditions that lead to those results. Different techniques highlight unique algebraic properties, ensuring equations are solved flexibly depending on their structure.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols. They form the building blocks of equations. In this task, expressions like \((x+3)^2\) and \(-10(x+5)\) needed expansion and simplification to form a cohesive equation. Simplifying such expressions is essential:

• Begin by expanding any products, like squares and multiplied terms, to eliminate parentheses.
• Combine like terms to streamline the expression into a more manageable form.
• Reorganize and express the simplified terms to reveal the equation's underlying structure, such as standard quadratic form.

Algebraic expressions teach critical manipulation skills necessary for higher-level math, reinforcing how variables and constants interact through arithmetic processes. Understanding and working with expressions is essential for solving equations, as it develops a strong foundation for recognizing patterns and fostering problem-solving skills.