Kitty lived next door to me, and we had been friends since childhood. We learned at the same school and studied

for me. We would walk together, sharing stories and laughing on the way.

One morning, Kitty knocked on my door as usual, but this time she seemed upset. She told me that she had made a mistake while solving a math problem and couldn"t figure out how to correct it. She asked me for help, so I invited her inside, and we sat down at the kitchen table to work on the problem together.

The problem Kitty was struggling with involved solving a quadratic equation. The equation was in the form \(ax^2 + bx + c = 0\), and she needed to find the values of \(x\) that satisfied the equation.

To solve this type of equation, we can use the quadratic formula: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\]

The quadratic formula gives us two possible values for \(x\), which are obtained by taking the sum or difference of the square root of \(b^2 - 4ac\), divided by \(2a\).

First, we need to identify the values of \(a\), \(b\), and \(c\) from the equation. Then we can substitute them into the quadratic formula and calculate the values of \(x\).

Let"s consider an example equation: \(2x^2 + 5x - 3 = 0\). In this equation, \(a = 2\), \(b = 5\), and \(c = -3\).

Using the quadratic formula, we can substitute these values into the formula and simplify the expression to find the values of \(x\):

\[x = \frac{{-5 \pm \sqrt{{5^2 - 4 \cdot 2 \cdot -3}}}}{{2 \cdot 2}}\]

Simplifying further:

\[x = \frac{{-5 \pm \sqrt{{25 + 24}}}}{{4}}\]
\[x = \frac{{-5 \pm \sqrt{{49}}}}{{4}}\]
\[x = \frac{{-5 \pm 7}}{{4}}\]

Now we have two possible values for \(x\):

\[x_1 = \frac{{-5 + 7}}{{4}} = \frac{2}{4} = \frac{1}{2}\]
\[x_2 = \frac{{-5 - 7}}{{4}} = \frac{-12}{4} = -3\]

Therefore, the solutions to the equation \(2x^2 + 5x - 3 = 0\) are \(x = \frac{1}{2}\) and \(x = -3\).

I explained to Kitty the step-by-step process of solving quadratic equations using the quadratic formula. We wrote down the formula, identified the values of \(a\), \(b\), and \(c\) from her equation, substituted the values into the formula, simplified the expression, and obtained the solutions. Kitty understood each step and gained confidence in solving similar problems in the future.

After successfully solving the problem, Kitty thanked me for the help and we continued our journey to school, chatting and laughing along the way.