Problem 20 - Algebra 2 and Trigonometry

Question

In \(9-26,\) solve each quadratic equation by completing the square. Express the answer in simplest radical form. $$ x^{2}-8 x+4=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 4 + 2\sqrt{3}\) and \(x = 4 - 2\sqrt{3}\).

Step 1: Move Constant to Right Side

Start by moving the constant term to the right side of the equation:\[x^2 - 8x = -4\]

Step 2: Complete the Square

To complete the square, take the coefficient of the linear term, divide it by 2, and square it.The linear term is \(-8\), so divide by 2 to get \(-4\), then square to get \(16\). Add \(16\) to both sides of the equation:\[x^2 - 8x + 16 = -4 + 16\]

Step 3: Simplify Equation

Simplify the right side of the equation:\[x^2 - 8x + 16 = 12\]

Step 4: Express as a Perfect Square

Write the left side as a perfect square trinomial:\[(x - 4)^2 = 12\]This step uses the fact that \((x-a)^2 = x^2 - 2ax + a^2\), where \(a = 4\).

Step 5: Solve for x

Take the square root of both sides and solve for \(x\). Remember to consider both the positive and negative roots:\[x - 4 = \pm \sqrt{12}\]Simplify \(\sqrt{12}\):\[\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\]So,\[x - 4 = \pm 2\sqrt{3}\]Finally, solve for \(x\):\[x = 4 \pm 2\sqrt{3}\]

Step 6: Write Final Answer

The solutions to the quadratic equation are:\[x = 4 + 2\sqrt{3} \quad \text{and} \quad x = 4 - 2\sqrt{3}\]

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form of \( ax^2 + bx + c = 0 \). Here, \(x\) represents the variable or unknown we are solving for, while \(a\), \(b\), and \(c\) are constants with \(a eq 0\). Quadratic equations are prevalent in various fields like physics and engineering, often used to model trajectories and other parabolic phenomena.

• The standard form is essential to recognize as it helps identify how to transform and manipulate the equation to solve it.
• Solving quadratic equations can be done using various methods, such as factoring, using the quadratic formula, or completing the square.

To solve the given quadratic equation using the method of completing the square, we convert it to a perfect square trinomial. This method involves making adjustments to represent the equation as a square of a binomial.

Simplest Radical Form

When solving equations that involve square roots, expressing results in simplest radical form means reducing square roots to their simplest equivalent. Simplifying square roots involves finding perfect square factors.

• Start by identifying the perfect square factor of the number inside the square root. For example, in \(\sqrt{12}\), notice that 4 (\(2^2\)) is the largest perfect square factor.
• Rewrite the expression as a product under the square root: \(\sqrt{12} = \sqrt{4 \times 3}\).
• Simplify by taking the square root of the perfect square factor: \(\sqrt{4} = 2\).
• The simplest form of \(\sqrt{12}\) becomes \(2\sqrt{3}\).

Simpler forms are preferred because they often provide clearer insight into the numbers and make further calculations easier. It is crucial when solving problems that require precision and simplicity in the final form.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that results from squaring a binomial. It can be recognized by its form \((x + a)^2 = x^2 + 2ax + a^2\). To transform a quadratic equation into a perfect square trinomial, we follow these steps:

• First, ensure the quadratic is in the form \(x^2 + bx\).
• Take the coefficient of the linear term (\(b\)), divide it by 2, and then square it. This completes the square.
• Add and subtract this square value within the equation to maintain equality.

In our example, the term was \(-8\). Dividing -8 by 2 gives -4, which squared results in 16. Thus, adding and subtracting 16 transforms the polynomial into a complete square \((x - 4)^2\). Transformations into perfect square trinomials simplify the equation, enabling easy solving by taking square roots.