91Ó°ÊÓ

The quadratic formula is a powerful tool for solving any quadratic equation, which is typically in the form \( ax^2 + bx + c = 0 \). This formula allows us to find the values of \( x \) that satisfy the equation, known as the roots of the equation. The formula is given by:

• \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

To use the quadratic formula, we must first identify the coefficients: \( a \), \( b \), and \( c \). These values come directly from the quadratic equation itself. In the example \( 4x^2 + 16x - 9 = 0 \), we have \( a = 4 \), \( b = 16 \), and \( c = -9 \).
Once the coefficients are identified, they are substituted into the quadratic formula to solve for \( x \). Notice the "\( \pm \)" symbol in the formula; it indicates that there are generally two possible values (solutions) for \( x \). This is because a quadratic equation represents a parabola which can intersect the x-axis at up to two points.

The discriminant is a part of the quadratic formula found under the square root symbol, represented by \( \Delta \). Mathematically, the discriminant is expressed as:

• \[ b^2 - 4ac \]

The value of the discriminant gives crucial information about the nature of the roots of the quadratic equation:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root (a repeated root).
• If \( \Delta < 0 \), the equation has no real roots (the roots are complex numbers).

In the example given, the discriminant calculates to \( 400 \). Since \( 400 > 0 \), it confirms that our quadratic equation \( 4x^2 + 16x - 9 = 0 \) has two distinct real solutions. Calculating the discriminant is a vital step, as it helps to determine how we interpret the resulting roots.

Real solutions refer to the values of \( x \) that result when the quadratic equation is solved using the quadratic formula and the discriminant is non-negative (i.e., \( \Delta \geq 0 \)). In the context of quadratic equations, these are the points where the curve of the graph crosses the x-axis.
Using the quadratic formula and the calculated discriminant \( \Delta = 400 \), we substitute the values back into the formula:

• \[ x = \frac{-16 \pm \sqrt{400}}{8} \]

Since \( \sqrt{400} = 20 \), we can further solve the equation to find our real solutions:

• For the positive root: \( x_1 = \frac{-16 + 20}{8} = \frac{1}{2} \)
• For the negative root: \( x_2 = \frac{-16 - 20}{8} = -\frac{9}{2} \)

These solutions \( \frac{1}{2} \) and \( -\frac{9}{2} \) are the real values of \( x \) that satisfy the quadratic equation. They provide the x-intercepts of the parabola represented by this specific equation.