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A quadratic equation is a type of polynomial equation that takes the form \(ax^2 + bx + c = 0\), where \(x\) is the variable, and \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The equation is called 'quadratic' because it includes \(x^2\), which is a term that represents a squared variable. Quadratic equations are essential in algebra and have a wide range of applications in various fields such as physics and engineering.
Examples of standard quadratic equations include:

• \(x^2 + 5x + 6 = 0\)
• \(2x^2 - 4x + 3 = 0\)

The solutions to a quadratic equation are the values of \(x\) that make the equation true. These solutions can be found using several methods like factoring, completing the square, and using the quadratic formula.

The discriminant is a part of the quadratic formula and is found inside the square root. It is denoted by \(\triangle = b^2 - 4ac\). The discriminant gives valuable information about the nature of the roots of the quadratic equation. Here are the key points:

• If the discriminant is greater than zero (\(\triangle > 0\)), the quadratic equation has two distinct real number solutions.
• If the discriminant is equal to zero (\(\triangle = 0\)), the quadratic equation has exactly one real number solution, also known as a repeated or double root.
• If the discriminant is less than zero (\(\triangle < 0\)), the quadratic equation has no real number solutions; instead, it has two complex solutions.

In our exercise, the discriminant is \(121\), which is greater than zero. This tells us that our equation \(x^2 + 3x - 28 = 0\) has two distinct real number solutions.

Real number solutions to a quadratic equation are the numerical values of \(x\) that satisfy the equation when substituted back into it. When the discriminant (\(b^2 - 4ac\)) is non-negative, the quadratic equation will have real number solutions. Specifically:

• Two distinct solutions if the discriminant is positive.
• One solution if the discriminant is zero.

For example, in our exercise, \(x^2 + 3x - 28 = 0\), we found the discriminant to be \(121\). Since \(121\) is positive, we proceed to find two distinct real number solutions for \(x\). These solutions were found to be \(x = 4\) and \(x = -7\). These values satisfy the original equation when substituted back.

Solving quadratic equations can be done using several methods: factoring, completing the square, and the quadratic formula. The quadratic formula is a universal method that works for all quadratic equations and is given by:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

• First, identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Calculate the discriminant using \(b^2 - 4ac\).
• Evaluate the square root of the discriminant.
• Substitute the values into the quadratic formula to find the solutions for \(x\).

In our exercise, we started with the equation \(x^2 + 3x - 28 = 0\). By identifying \(a = 1\), \(b = 3\), and \(c = -28\), and calculating the discriminant as \(121\), we applied the quadratic formula to find \(x = 4\) and \(x = -7\). These are the solutions to the quadratic equation.