Problem 67 - Algebra 1: Concepts and Skills

Question

Choose a method and solve the quadratic equation. Explain your choice. $$ x^{2}-6 x+7=0 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(x^{2}-6 x+7=0\) are \(x = 3 + \sqrt{2}\) and \(x = 3 - \sqrt{2}\).

Step 1: Identify the coefficients

The coefficients from the quadratic equation \(x^{2}-6 x+7=0\) are: a = 1, b = -6, and c = 7.

Step 2: Apply the Quadratic Formula

The Quadratic Formula is given by \((-b ± \sqrt {b^{2}-4ac})/(2a)\). Substituting the coefficients \(a = 1\), \(b = -6\) and \(c = 7\) into this formula gives \((6 ± \sqrt {(-6)^{2}-4*1*7})/(2*1)\).

Step 3: Simplify the expression

Simplifying under the radical, gives \(6 ± \sqrt{36 - 28}\) / 2. Continue = \(6 ± \sqrt{8}\) / 2. We see now that the square root of 8 simplifies to \(2*\sqrt{2}\), so substituting gives \(6 ± 2 * \sqrt{2}\) / 2.

Step 4: Solve for x

Now evaluating this expression gives for x two possible solutions \(x = 3 + \sqrt{2}\) and \(x = 3 - \sqrt{2}\).

Key Concepts

Quadratic Formula

The Quadratic Formula is an essential tool for solving quadratic equations. Quadratic equations are polynomial equations with the highest exponent of 2 and generally follow the form: \[ ax^2 + bx + c = 0 \] The coefficients \(a\), \(b\), and \(c\) are real numbers, where \(a eq 0\). If \(a\) is zero, the equation becomes linear, not quadratic.
The Quadratic Formula allows us to find the roots of the equation by substituting these coefficients into the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
The part of the formula under the square root, \(b^2 - 4ac\), is called the discriminant, and it determines the nature of the roots. Use the Quadratic Formula especially when factoring the quadratic is challenging or if the roots are not easily recognizable integers.

Roots of Quadratics

Finding the roots of a quadratic equation means determining the values of \(x\) that satisfy the equation. These are also known as the solutions to the equation.
There are various methods for finding these roots including:

• Factoring
• Completing the square
• Using the Quadratic Formula

The choice of method often depends on the specific form of the quadratic and the ease of solution.
In our example, the roots were found using the Quadratic Formula, leading to two solutions:

• \(x = 3 + \sqrt{2}\)
• \(x = 3 - \sqrt{2}\)

These solutions can be real, imaginary, rational, or irrational, depending on the discriminant \(b^2 - 4ac\). When the discriminant is positive, like 8 in our example, there are two distinct real roots.

Simplifying Radicals

Understanding how to simplify radicals is crucial when using the Quadratic Formula as many solutions involve square roots.
A radical expression is fully simplified when there are no perfect square factors other than 1 inside the radical sign and no fractions under the square root. The process of simplification involves expressing the number under the root as a product of a perfect square and another factor.
In our example, \(\sqrt{8}\) was simplified by recognizing that \(8\) can be broken down into \(4\times 2\), which makes \(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2}\). Since \(\sqrt{4} = 2\), this simplifies further to \(2\sqrt{2}\).
Such simplifications are important for expressing solutions in their simplest form, making them easier to understand and work with.