Problem 49 - Algebra 1: Concepts and Skills

Question

Solve the quadratic equation. $$ x^{2}+8 x-6=0 $$

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \( x = -4 + \sqrt{22} \) and \( x = -4 - \sqrt{22} \).

Step 1: Identify a, b and c

In the quadratic equation, assign \( a = 1 \), \( b = 8 \), and \( c = -6 \). These values will be substituted into the Quadratic Formula.

Step 2: Apply the Quadratic Formula

Insert values of \( a \), \( b \), and \( c \) into the Quadratic Formula which is \( x = [-b \pm \sqrt{b^2 - 4ac}] / 2a \). Doing so, yields: \( x = [-8 \pm \sqrt{8^2 - 4*1*(-6)}] / 2*1 = [-8 \pm \sqrt{64 + 24}] / 2 \).

Step 3: Simplify the result

Simplify underneath the square root: \( x = [-8 \pm \sqrt{88}] / 2 \), which simplifies to \( x = -4 \pm \sqrt{22} \).

Key Concepts

Quadratic Formula

Quadratic equations can be daunting, but the Quadratic Formula is one of the most powerful tools at our disposal for solving them. This formula provides a straightforward method for finding the roots of any quadratic equation, which has the general form of ax^2 + bx + c = 0. The roots are the values of x that make the equation true, and the Quadratic Formula states that these roots can be found using the expression x = [-b ± sqrt(b^2 - 4ac)] / 2a.

To use it, simply identify the coefficients a, b, and c from your quadratic equation and plug them into the formula. Once you perform the calculations including the square root and the division, you will end up with two solutions, also known as the roots. These can be real numbers or complex numbers, depending on whether the expression under the square root sign, called the discriminant, is positive or negative.

Factoring Quadratics

When a quadratic equation is written in the form ax^2 + bx + c = 0 and can be factored, it's like cracking a code to reveal simpler algebraic expressions that can be solved with ease. Factoring quadratics involves finding two binomials that, when multiplied together, give us the original quadratic equation. This method is highly effective when the equation is factorable, meaning the equation can be expressed as (dx + e)(fx + g) = 0, where dx * fx = a, eg = c, and df + eg = b.

If the quadratic does factor neatly, then setting each binomial equal to zero and solving for x gives the roots of the quadratic equation. This method often requires some trial and error, as well as a good understanding of factors and multiplication, but it's a powerful approach when applicable.

Roots of Polynomials

Understanding the roots of polynomials, particularly quadratic equations, is essential for solving many algebraic problems. The roots are the solutions to the equation — the values of x where the polynomial equals zero. They are the points where the graph of the polynomial crosses the x-axis. The process of finding these roots may vary depending on the structure of the polynomial.

For quadratics, which are polynomials of degree 2, there can be two roots, one root, or no real roots at all. The number of real roots is determined by the discriminant (b^2 - 4ac). A positive discriminant indicates two distinct real roots, a discriminant of zero indicates a single, repeated real root, and a negative discriminant signals that there are two complex roots.

Algebraic Expressions

Algebraic expressions play a key role in formulating and solving mathematical problems. They are combinations of letters (variables), numbers, and operations that represent mathematical relationships. The expression for a quadratic equation, ax^2 + bx + c, is a prime example, with a, b, and c being constants that determine the specific shape and position of the parabola it represents.

Manipulating algebraic expressions is essential for solving equations. Techniques such as combining like terms, factoring, expanding, and using the distributive property help us rewrite these expressions in ways that are easier to work with, ultimately leading to the solution of the equation at hand.