Question

Write the quadratic equation in standard form. Solve using the quadratic formula. $$-16 b=-8 b^{2}-8$$

Short answer

The solutions to the quadratic equation are both \(b=-1\).

Step-by-step solution

Write the equation in standard form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Rearrange the equation inside the exercise from \(-16 b = -8 b^{2}-8\) to \(8 b^{2} + 16 b + 8 = 0\). However, all terms can be divided by 8 for simplification. Now, the equation become \( b^2 + 2b + 1 = 0\).

Calculate the discriminant

To use the quadratic formula, first calculate the discriminant, which is \(b^{2} - 4ac\). In this equation, \(a=1\), \(b=2\), and \(c=1\). Thus, the discriminant equals \(2^2 - 4*1*1 = 4 - 4 = 0\). The discriminant is also the value inside the square root in the quadratic formula.

Solve the quadratic equation

Once the discriminant has been found, plug the values into the quadratic formula to find the solution: \(b = \frac{-2 \pm \sqrt{0}}{2 * 1}\). Simplifying this, we get \(b = -1, -1 = [-1,-1]\).

Key concepts

Solving Quadratic Equations

A quadratic equation is a second-degree polynomial equation in a single variable x, with a nonzero coefficient for the x^2 term. It has the general form \(ax^2 + bx + c = 0\), where 'a' is not equal to 0. To solve a quadratic equation means to find the values of x that make the equation equal to zero. There are various methods to solve quadratic equations, including factoring, completing the square, using the quadratic formula, and graphing.

In the example exercise, the quadratic equation is first written in the standard form by rearranging and simplifying the given equation, resulting in \(b^2 + 2b + 1 = 0\). To solve for b, one could factor this equation or apply the quadratic formula, which is often the most straightforward method, especially when factoring is complex or impossible.

Quadratic Formula

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is derived from the process of completing the square and can solve any quadratic equation. It provides the solution(s) for x by substituting the coefficients 'a', 'b', and 'c' from the standard form of the quadratic equation.

The symbol '\(\pm\)' indicates there are usually two solutions: one where you add the square root of the discriminant and one where you subtract it. However, if the discriminant equals zero, there is only one real solution, as it is the case in the example where we get two identical solutions for b, which is -1. This solution implies that the parabola that represents the quadratic equation merely touches the x-axis at a single point, which also suggests that the quadratic has a perfect square factorization.

Discriminant of a Quadratic Equation

The discriminant in a quadratic equation is the part under the square root in the quadratic formula, \(b^2 - 4ac\), and it determines the number and type of solutions. If the discriminant is greater than zero, there are two distinct real solutions. If it is exactly zero, there is one real solution, often referred to as a repeated or double root. Lastly, if the discriminant is less than zero, there are two complex solutions.

In our example, after simplifying the standard form we calculated the discriminant, which was zero \(2^2 - 4 \times 1 \times 1 = 4 - 4 = 0\). This means the quadratic equation \(b^2 + 2b + 1 = 0\) has one unique real solution, which we found to be b = -1, after using the quadratic formula. The discriminant not only provided us with the types of solutions but also indicated that the quadratic could be factored into \(b + 1)^2 = 0\), confirming that b = -1 is indeed a double root.