91Ó°ÊÓ

The quadratic formula is a powerful tool for finding the roots, or zeros, of any quadratic equation in the form of \(ax^2 + bx + c = 0\). It looks like this:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
This formula gives you the solutions to the equation by substituting the coefficients \(a\), \(b\), and \(c\). These coefficients are the numbers in front of the \(x^2\), \(x\), and constant term in the quadratic equation. To use the quadratic formula, follow these steps easily:

• Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Calculate the discriminant, which is \(b^2 - 4ac\).
• Substitute \(a\), \(b\), and the discriminant back into the quadratic formula.
• Simplify to find the values of \(x\).

The discriminant is a part of the quadratic formula found inside the square root (\( b^2 - 4ac \)). It tells you whether the quadratic equation has zero, one, or two real solutions. Here's how it works:

• If the discriminant is positive (\( > 0 \)), the quadratic equation has two distinct real solutions.
• If the discriminant is zero (\( = 0 \)), the quadratic equation has exactly one real solution.
• If the discriminant is negative (\( < 0 \)), the quadratic equation has no real solutions, but two imaginary solutions.

In our example, \( 4x^2 - 4x - 5 \), the discriminant is \( 96 \) (since \( (-4)^2 - 4(4)(-5) = 16 + 80 \)). Since 96 is positive, we have two distinct real solutions for this quadratic equation.

To find the zeros of a quadratic function algebraically means to solve the equation by using algebraic methods like the quadratic formula. Here's an example of how to do it for the function \( f(x) = 4x^2 - 4x - 5 \).

Step-by-step:
1. Set up the equation \( f(x) = 0 \):
\( 4x^2 - 4x - 5 = 0 \).
2. Identify \(a\), \(b\), and \(c\) (in this case, \(a = 4\), \(b = -4\), \(c = -5\)).
3. Calculate the discriminant:
\( (-4)^2 - 4(4)(-5) = 96 \).
4. Substitute back into the quadratic formula:
\( x = \frac{-(-4) \pm \sqrt{96}}{2(4)} \).
5. Simplify the expression:
\( x = \frac{4 \pm 4\sqrt{6}}{8} = \frac{1 \pm \sqrt{6}}{2} \).
The zeros of the function \( f(x) \) are \( x = \frac{1 + \sqrt{6}}{2} \) and \( x = \frac{1 - \sqrt{6}}{2} \).