3.3. Quadratic Formula

The quadratic formula is given by:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
This formula allows us to find the roots of any quadratic equation of the form $ax^2 + bx + c = 0$, where $a \ne 0$.

The quadratic formula provides a straightforward way to solve quadratic equations. It uses the coefficients of the equation:

• a is the coefficient of $x^2$.
• b is the coefficient of $x$.
• c is the constant term. The expression under the square root, $b^2 - 4ac$, is called the discriminant and plays an important role in determining the nature and number of roots.

Think of the quadratic formula as a recipe that tells you how to mix certain ingredients (the coefficients a, b, and c) to find the answer (the roots) of a quadratic equation. Just like a recipe can help you bake a cake with a specific height (the roots), the quadratic formula helps you find where the parabola intersects the x-axis.

The discriminant $\Delta = b^2 - 4ac$ determines the number and type of roots:

• If $\Delta > 0$: Two real roots
• If $\Delta = 0$: One real root
• If $\Delta < 0$: No real roots (complex solutions)

The discriminant gives important information about the roots of the quadratic equation.

• Two real roots occur when the discriminant is positive, which means the parabola crosses the x-axis at two distinct points.
• One real root happens when the discriminant is zero, indicating the vertex of the parabola touches the x-axis at one point.
• No real roots means the parabola does not intersect the x-axis at all and the solutions are complex numbers instead.

Imagine you’re determining whether a toy will bounce or not when dropped. If the initial position and force allow it to bounce off the ground twice, you have two real roots. If it only barely touches the ground before settling, that’s like one real root. If it simply doesn’t bounce and goes straight down, you have no real roots!

To solve the equation $2x^2 - 4x - 6 = 0$ using the quadratic formula:
Identify $a = 2$, $b = -4$, $c = -6$:
$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(-6)}}{2(2)}$
$x = \frac{4 \pm \sqrt{16 + 48}}{4} = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}$
Thus, $x = 3$ or $x = -1$.

In this example, we follow the steps of using the quadratic formula to find the roots of a specific quadratic equation.

1. First, we identify the values of a, b, and c from the equation.
2. We substitute these values into the quadratic formula.
3. Next, we calculate the discriminant and simplify the expression.
4. Finally, we will have two potential values of x, which represent the solutions to the quadratic equation.

Imagine you're trying to find out the time it takes for a ball to hit the ground when it's thrown upwards with a specific force. Each value you calculate (root) can represent a moment when the ball is at a certain point above or below the ground. The solutions you find give you insights on how long it takes before the ball hits the ground.

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