91Ó°ÊÓ

Quadratic equations are polynomial equations of the form \[ ax^2 + bx + c = 0 \], where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. To solve a quadratic equation, you need to find values of \( x \) that make the equation true. One of the most reliable methods for solving these equations is the quadratic formula. The quadratic formula is given by:
\[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation.

Follow these steps to solve using the quadratic formula:

• Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
• Insert these values into the quadratic formula.
• Calculate the discriminant (the value under the square root sign).
• Solve for both possible answers by using \( + \) and \( - \) in the formula.

By following these steps, you can find the exact solutions to any quadratic equation.

The discriminant is an important part of the quadratic formula. It is represented as \( b^2 - 4ac \). The value of the discriminant helps us understand the nature of the roots of the quadratic equation:

* When the discriminant is positive (\( b^2 - 4ac > 0 \)), the quadratic equation has two distinct real solutions.

* When the discriminant is zero (\( b^2 - 4ac = 0 \)), the quadratic equation has exactly one real solution. This means the parabola touches the x-axis at one point.

* When the discriminant is negative (\( b^2 - 4ac < 0 \)), the quadratic equation has no real solutions. Instead, it has two complex solutions.

In the given problem, the equation is \( 5m^2 + 3m - 2 = 0 \), and we found that the discriminant is 49 (which is positive). Therefore, there are two distinct real solutions.

Coefficients in a quadratic equation are the numbers in front of the variables in the standard form \( ax^2 + bx + c = 0 \). These coefficients (\( a \), \( b \), and \( c \)) determine the shape and position of the parabola represented by the equation.

Here is what each coefficient represents:

• \( a \) (the coefficient of \( x^2 \)) determines the direction and width of the parabola. If \( a \) is positive, the parabola opens upwards. If \( a \) is negative, it opens downwards. A larger absolute value of \( a \) makes the parabola narrower, while a smaller absolute value makes it wider.
• \( b \) (the coefficient of \( x \)) affects the position of the vertex of the parabola along the x-axis.
• \( c \) (the constant term) moves the parabola up or down along the y-axis.

In the exercise \( 5m^2 + 3m - 2 = 0 \), the coefficients are \( a = 5 \), \( b = 3 \), and \( c = -2 \). Knowing these coefficients is crucial for applying the quadratic formula and understanding the properties of the quadratic equation.