Problem 79 - Algebra 1: Concepts and Skills

Question

Solve \(x^{2}+8 x-2=0\). A. \(-4 \pm 3 \sqrt{2}\) B. \(-4 \pm 2 \sqrt{2}\) C. \(4 \pm 3 \sqrt{2}\) D. \(4 \pm \sqrt{16}\)

Step-by-Step Solution

Answer

The solution to the quadratic equation is \(x=-4 \pm 3\sqrt{2}\), so the answer is (A) \(x=-4 \pm 3\sqrt{2}\).

Step 1: Identification

Identify the values of a, b, and c in the given quadratic equation. In the equation \(x^{2}+8 x-2=0\), a=1, b=8 and c=-2.

Step 2: Plug into Quadratic Formula

Substitute the values of a, b, and c into the quadratic formula \(x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}\). This gives \(x=\frac{-8\pm \sqrt{8^{2}-4(1)(-2)}}{2(1)}\).

Step 3: Solve the Equation

Solve the equation to find the values of x. This simplifies to \(x=\frac{-8\pm \sqrt{64+8}}{2}\), which further simplifies to \(x=\frac{-8\pm \sqrt{72}}{2}\). This can also be written as \(x=-4 \pm \sqrt{18}\), which simplifies to \(x=-4 \pm 3\sqrt{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions or 'roots' of a quadratic equation. It's derived from completing the square in a general quadratic equation of the form \( ax^{2} + bx + c = 0 \). Applying the quadratic formula, \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \), can quickly yield the two possible values for \( x \) that solve the equation.

The formula consists of several components:

• The term \( -b \) on the numerator indicates that we are considering the opposite sign of the coefficient\( b \).
• The \( \pm \) symbol signifies that there are typically two solutions for \( x \): one through addition and one through subtraction.
• The term \( b^{2} - 4ac \) under the square root is known as the discriminant. Its value can determine the nature and number of roots.
• The denominator \( 2a \) divides the entire expression to solve for the x-intercepts of the parabola represented by the quadratic equation.

When applying the quadratic formula to an equation, it's crucial to accurately identify the values of coefficients \( a \), \( b \), and \( c \) and then carefully perform the arithmetic to arrive at the correct roots. The quadratic formula is universally applicable to all quadratic equations, making it an essential method in algebra.

Factoring Quadratics

Factoring quadratics involves rewriting the quadratic equation as a product of its factors. It's based on the principle that if \( ax^{2} + bx + c = 0 \), then it can be expressed as \( (mx + n)(px + q) = 0 \) where \( m, n, p, \text{and} \ q \) are numbers that when multiplied, give \( a \) and \( c \), and when added, give \( b \).

The process of factoring can vary depending on the equation and might involve techniques such as:

• Finding a greatest common factor (GCF) if there is one.
• Applying the difference of squares if applicable.
• Trial and error or the 'ac method' for tricker cases.

If the quadratic can be factored, it provides a straightforward way of finding the roots. By setting each factor equal to zero and solving for \( x \), we find the solutions to the equation. It's important to note that factoring is not always possible, especially if the equation does not simplify to a product of integer factors. In such cases, the quadratic formula is a fail-proof alternative.

Roots of Polynomial

The roots of a polynomial, also known as zeros, are the values that make the polynomial equal to zero. For quadratic equations, which are polynomials of degree 2, there can be up to two real roots. Finding these roots is fundamental in understanding the graph and behavior of the polynomial function.

There are three main cases based on the discriminant \( (b^{2} - 4ac) \) related to the roots:

• If the discriminant is positive, there will be two distinct real roots.
• If it is zero, there will be exactly one real root (also known as a repeated or double root).
• If it is negative, there will be no real roots, but rather a pair of complex roots.

In the context of a quadratic equation, these roots correspond to the points where the graph of the equation intersects the x-axis. They are essential in sketching parabolas and solving a variety of application problems. Whether through factoring, completing the square, or using the quadratic formula, identifying the roots helps us understand the structure and solutions of the equation.