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To solve a quadratic equation, one of the most common methods is using the quadratic formula. The quadratic formula is a powerful tool because it gives a direct way to find the solutions (roots) of any quadratic equation in the form of \(ax^2 + bx + c = 0\). This is particularly useful because not all quadratic equations can be factored easily.

Here's a step-by-step guide to using the quadratic formula:

• Identify the coefficients: For \(ax^2 + bx + c = 0\), determine the values of \a, b,\ and \c.\
• Write down the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\. \]
• Substitute the coefficients into the formula.
• Simplify the expression under the square root (called the discriminant).
• Solve for \x\ by performing the arithmetic operations indicated by the formula.

If the quadratic formula seems complicated, don't worry. With practice, it becomes second nature!

Let's consider the given example: \(x^2 + 3x - 4 = 0\)

• Identify coefficients: \(a = 1, b = 3, c = -4\)
• Write quadratic formula: \[ x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-4)}}{2(1)} \]
• Simplify it step-by-step to find the roots.

The value of the discriminant tells us a lot about the solutions of the quadratic equation:

• If \(\bigtriangleup > 0\), there are two distinct real solutions.
• If \(\bigtriangleup = 0\), there is exactly one real solution (a repeated root).
• If \(\bigtriangleup < 0\), the solutions are complex or imaginary (not real).

Let's apply this to our example: \(\bigtriangleup = 3^2 - 4(1)(-4) = 9 + 16 = 25\)

Since \(\bigtriangleup = 25\), which is greater than zero, we know that there are two distinct real roots for the quadratic equation \({\textstyle x^2 + 3x - 4 = 0}\).

The discriminant not only aids in solving the quadratic equation but also provides insight into how the solutions are structured.

The solutions or roots of a quadratic equation are the values of \x\ that satisfy the equation. These roots can be real or complex, depending on the value of the discriminant.

Using the quadratic formula \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \, \] we get two roots: one by adding the square root of the discriminant and one by subtracting it.

In our example, we found \[ x = \frac{-3 \pm \sqrt{25}}{2} \]. This led to:

• First root: \({\textstyle x = \frac{-3 + 5}{2} = 1}\)
• Second root: \({\textstyle x = \frac{-3 - 5}{2} = -4}\)

Therefore, the quadratic equation \(x^2 + 3x - 4 = 0\) has roots \(x = 1\) and \(x = -4\).

Roots can help us understand different characteristics of the quadratic equation. For example:

• The sum of the roots is given by \(-b/a\).
• The product of the roots is given by \(c/a\).

In this case, the sum is \({-3/1} = -3\) and the product is \({-4/1} = -4\). Knowing the roots allows for greater insight into the behavior of the quadratic equation's graph.