Problem 11 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation by completing the square. $$ x^{2}+10 x-10=0 $$

Step-by-Step Solution

Answer

The solutions for \( x \) are \( x_1 = -5 + \sqrt{35} \), \( x_2 = -5 - \sqrt{35} \)

Step 1: Making The Coefficient of \( x^2 \) Equal to 1

Here, in our given equation \( x^{2}+10x-10=0 \), the coefficient of \( x^2 \) is already 1. So, we can proceed to the next step.

Step 2: Rearrange the Equation

The idea behind completing the square is to write the quadratic equation in the form \( (x + a)^2 = b \). for that we group the \( x's \) together and the constant on the other side: \( x^2 + 10x = 10 \).

Step 3: Completing the Square

Now we want to add a constant (let's call it \( p \)) on both sides of the equation so that the left side becomes a perfect square. That constant \( p \) will be \( \left(\frac{b}{2}\right)^2 \), where \( b \) is the coefficient of \( x \). Here, \( b = 10 \), so \( p = \left(\frac{10}{2}\right)^2 = 25 \). Adding \( p \) on both sides: \( x^2 + 10x + 25 = 10 + 25 \). The equation becomes \( (x + 5)^2 = 35 \).

Step 4: Solving for \( x \)

Taking square root on both sides of the equation, we get: \( x + 5 = \pm\sqrt{35} \). So, the solutions for \( x \) are: \( x_1 = -5 + \sqrt{35} \) and \( x_2 = -5 - \sqrt{35} \).

Key Concepts

Quadratic Equations

Quadratic equations are a staple topic in algebra and appear in various forms. A standard quadratic equation is written as \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The variable \( x \) represents the unknown value we aim to solve for.
Quadratic equations often have two solutions because they are second-degree polynomials - meaning, they have the highest exponent of 2. These equations can be solved using different methods such as factoring, completing the square, or using the quadratic formula. Knowing when and how to use each method is crucial for solving problems efficiently.
A quadratic equation can depict a parabola when graphed. The solutions provide the values at which the graph intersects the x-axis, also referred to as the roots of the equation. Recognizing the characteristics of quadratic equations helps in understanding the solutions’ behavior and the nature of the graph.

Perfect Square Trinomials

Perfect square trinomials are specific quadratic expressions that can be rewritten as the square of a binomial. An expression of the form \( x^2 + 2ax + a^2 \) is a perfect square trinomial, equaling \((x + a)^2\). This concept is essential when completing the square.
To identify or construct a perfect square trinomial, you will:

• Determine the middle term's coefficient, often labeled as \( b \) in a quadratic equation \( x^2 + bx + c \).
• Take half of the middle term's coefficient and square it to obtain the constant term \( a^2 \).
• Add \( a^2 \) to both sides of the equation to transform it into a perfect square trinomial.

Understanding perfect square trinomials aids in simplifying and solving quadratic equations, as it allows us to replace complex expressions with easier-to-manage binomials.

Solving Equations

Solving equations, particularly quadratic equations, involves finding the values of the variable that satisfy the equation. In the case of quadratic equations, we want to discover the unknown value \( x \) such that \( ax^2 + bx + c = 0 \) holds true.
Using the method of completing the square, we convert a quadratic equation into a perfect square form, \((x + a)^2 = b\). This makes it relatively straightforward to solve for \( x \) by taking the square root of both sides and then isolating \( x \).
This method is particularly useful when the quadratic equation is not easily factored. However, ensure you perform all steps, such as adding the same constant to both sides, to maintain the equation's balance. Finally, checking the solutions in the original equation verifies the accuracy of your work.
Practicing different methods and understanding their applications enhance proficiency in solving equations efficiently.