91Ó°ÊÓ

Quadratic equations are polynomial equations of degree 2. They have the general form \(ax^2 + bx + c = 0\). The graph of a quadratic equation is a parabola. Quadratic equations often appear in physics and engineering, so understanding them is essential.
To solve a quadratic equation and find its x-intercepts, you'll need to set the equation to zero and find the values of x that satisfy it.
These x-intercepts are the solutions where the graph of the quadratic equation crosses the x-axis. In our example, the quadratic equation is \(2x^2 - 3x - 1 = 0 \). Here, \(a = 2\), \(b = -3\), and \(c = -1\). Identifying these coefficients is crucial for using the quadratic formula in the next steps.

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula allows you to find the x-intercepts of the quadratic equation by substituting the coefficients a, b, and c. In our example \(2x^2 - 3x - 1 = 0\), substituting \(a = 2\), \(b = -3\), and \(c = -1\) into the quadratic formula gives us:
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-1)}}{2(2)} \]
It's essential to perform the calculations step by step to avoid errors. Simplify inside the square root (radical) first:
\[ (-3)^2 = 9 \], \[ 4(2)(-1) = -8 \]
Adding these results:
\[ 9 + 8 = 17 \]
Now the expression under the square root becomes \( \sqrt{17} \).

Once we have simplified under the square root, the next steps involve simplifying the rest of the quadratic formula to get the solutions for x. Substituting back we have:
\[ x = \frac{3 \pm \sqrt{17}}{4} \]
Splitting it into two parts:
\[ x = \frac{3 + \sqrt{17}}{4} \]
and
\[ x = \frac{3 - \sqrt{17}}{4} \]
These two solutions represent the x-intercepts of the quadratic equation \(2x^2 - 3x - 1 = 0\). To summarize:

• Quadratic equations are solved by setting them to zero.
• Coefficients (a, b, c) are identified.
• The quadratic formula is used to find the x-intercepts.
Understanding these steps ensures you're prepared to tackle any quadratic equation you encounter in your studies.
Practice more to get comfortable with each step and always double-check your work.