The Quadratic Formula: Videos & Practice Problems

The quadratic formula is a powerful tool for solving any quadratic equation, which is typically expressed in the standard form \( ax^2 + bx + c = 0 \). The formula is given by:

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

To effectively use the quadratic formula, it is essential to identify the coefficients \( a \), \( b \), and \( c \) from the equation. The term \( b^2 - 4ac \) is known as the discriminant, and it helps determine the nature of the roots. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, the roots are complex.

For example, consider the equation \( x^2 + 2x - 3 = 0 \). Here, \( a = 1 \), \( b = 2 \), and \( c = -3 \). Plugging these values into the quadratic formula yields:

\( x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \)

Calculating the discriminant gives \( 2^2 - 4 \cdot 1 \cdot (-3) = 4 + 12 = 16 \). Thus, the equation simplifies to:

\( x = \frac{-2 \pm 4}{2} \)

This results in two solutions: \( x = 1 \) and \( x = -3 \).

In another example, for the equation \( x^2 - 5x = -1 \), we first rearrange it to standard form, resulting in \( x^2 - 5x + 1 = 0 \). Here, \( a = 1 \), \( b = -5 \), and \( c = 1 \). Substituting these values into the quadratic formula gives:

\( x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \)

Calculating the discriminant yields \( 25 - 4 = 21 \), leading to:

\( x = \frac{5 \pm \sqrt{21}}{2} \)

This expression represents two solutions: \( x = \frac{5 + \sqrt{21}}{2} \) and \( x = \frac{5 - \sqrt{21}}{2} \).

In summary, the quadratic formula is a versatile method for solving quadratic equations, applicable regardless of whether the equation can be factored easily. Mastery of this formula, along with practice in identifying coefficients and simplifying expressions, will enhance your problem-solving skills in algebra.

The discriminant is a crucial concept in understanding the nature of solutions for quadratic equations. It is defined as the expression under the square root in the quadratic formula, represented mathematically as \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation in the standard form \(ax^2 + bx + c = 0\).

The value of the discriminant provides insight into the number and type of solutions the quadratic equation has:

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, often referred to as a repeated or double root.
• If \(D < 0\), the equation has no real solutions, but instead has two complex (imaginary) solutions.

To illustrate how to use the discriminant, consider the following examples:

1. For the quadratic equation \(2x^2 + 3x - 2 = 0\), we identify \(a = 2\), \(b = 3\), and \(c = -2\). Plugging these values into the discriminant formula gives:

Since \(D = 25\) (which is positive), this equation has two real solutions.

2. In the case of \(4x^2 + x + 2 = 0\), we have \(a = 4\), \(b = 1\), and \(c = 2\). The discriminant calculation is:

Here, \(D = -31\) (negative), indicating that there are no real solutions, only two imaginary solutions.

3. Finally, for the equation \(x^2 - 10x + 25 = 0\), we find \(a = 1\), \(b = -10\), and \(c = 25\). The discriminant is calculated as follows:

With \(D = 0\), this equation has one real solution.

Understanding the discriminant allows for quick assessments of quadratic equations without the need for complete solutions, making it a valuable tool in algebra.