Problem 48 - Algebra 1

Question

Write the quadratic equation in standard form. Solve using the quadratic formula. $$2 q^{2}-6=-4 q$$

Step-by-Step Solution

Answer

The given equation in standard form is \(2q^2 + 4q + 6 = 0\). However, there are no real solutions as the discriminant is negative.

Step 1: Writing the equation in standard form

The given equation is \(2q^2 - 6 = -4q\). The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). To convert the given equation into standard form, we'll need to get all terms on one side of the equation: \(2q^2 + 4q + 6 = 0\)

Step 2: Solving using the Quadratic Formula

The quadratic formula is given by \[x = (-b ± √(b^2 - 4ac)) / (2a)\]. Let's substitute \(a=2\), \(b=4\), and \(c=6\) into the quadratic formula. This gives \[ q = (-4 ± √((4^2) - 4*2*6)) / (2*2) \].

Step 3: Simplify the Equation

Now we simplify under the square root: \[ q = (-4 ± √(16 - 48)) / 4 \]. Since the number under the square root (known as the discriminant) is negative, this equation has no real solutions.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the second degree. A quadratic equation takes the form of \[ ax^2 + bx + c = 0 \], where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The quadratic formula helps find the values of \( x \) that satisfy this equation.
The quadratic formula is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
- Here, \( b^2 - 4ac \) is termed the discriminant.- The two solutions provided by the formula are due to the "+" and "-" signs, representing the possibility of two values for \( x \).
This formula is essential because it offers a straightforward method to find the roots of any quadratic equation, provided that a mathematical solution exists for the equation.

Standard Form

Writing a quadratic equation in standard form is key to easily solving it, especially when using methods like the quadratic formula. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \].
- Here, \( a \), \( b \), and \( c \) are real numbers, where \( a eq 0 \).- This equation is called 'quadratic' because the highest degree term is squared (\( x^2 \)).
To rewrite an equation in standard form, make sure all terms are on one side of the equation and the equation is set to zero. For instance, with \[ 2q^2 - 6 = -4q \], we need to bring all terms to one side:\[ 2q^2 + 4q + 6 = 0 \].Setting it to zero allows us to apply solving techniques like the quadratic formula or factoring, which rely on the equation being in this form.

Discriminant

The discriminant is a key part of the quadratic formula and plays a crucial role in determining the nature of the solutions of a quadratic equation.
It is represented within the formula as \( b^2 - 4ac \).
- The discriminant helps understand how many real roots a quadratic equation has:

• If \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real solutions.
• If \( b^2 - 4ac = 0 \), there is exactly one real solution, and it is often called a repeated or "double" root.
• If \( b^2 - 4ac < 0 \), the quadratic equation has no real solutions, resulting in complex or imaginary solutions.

In our example, the discriminant is negative \( (16 - 48 = -32) \). This indicates that the quadratic equation \( 2q^2 + 4q + 6 = 0 \) has no real roots, meaning it cannot be solved within the real number system.