91Ó°ÊÓ

To effectively solve a quadratic equation using the quadratic formula, it is crucial to have the equation in its standard form. This form is expressed as \( ax^2 + bx + c = 0 \). Each of the terms in this equation has a specific role:

• \( a \): the coefficient of \( x^2 \), representing the quadratic term.
• \( b \): the coefficient of \( x \), known as the linear term.
• \( c \): the constant term without an \( x \) attached.

For instance, when we look at the initial equation \( 5x^2 + 7x = 3 \), it's not in the standard form yet because it doesn't equal zero. So, we need to move all terms to one side of the equation by subtracting 3 from both sides, which gives us \( 5x^2 + 7x - 3 = 0 \). Now that we have the equation in standard form, we can identify the values of \( a \), \( b \), and \( c \) as 5, 7, and -3, respectively. This preparation is essential for applying the quadratic formula correctly.

The discriminant is a part of the quadratic formula that appears under the square root sign. In mathematical terms, it is expressed as \( b^2 - 4ac \). This value is pivotal as it determines the nature and number of roots for the quadratic equation.

• If the discriminant is positive, there are two distinct real roots.
• If it equals zero, it results in one real root, often referred to as a repeated or double root.
• If the discriminant is negative, the equation has no real roots, but two complex roots.

For our specific problem, by substituting the values \( a = 5 \), \( b = 7 \), and \( c = -3 \), we compute the discriminant as \( 49 + 60 = 109 \). Since 109 is positive, this indicates our quadratic equation will have two distinct real roots. Understanding this concept highlights whether or not real solutions will exist without actually solving the equation.

The final destination when using the quadratic formula is finding the roots or solutions of the quadratic equation. The formula given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) directly calculates these roots. Here, the sign \( \pm \) suggests that there are two solutions.

• This means one root is found by using the plus sign: \( x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \)
• The other is found by using the minus sign: \( x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \)

In our case, after calculating \( \sqrt{109} \) and substituting back into our formula, we find \( x_1 = \frac{-7 + \sqrt{109}}{10} \) and \( x_2 = \frac{-7 - \sqrt{109}}{10} \). These represent the roots of the equation \( 5x^2 + 7x - 3 = 0 \). The quadratic formula is incredibly powerful as it provides both roots directly from the coefficients of the quadratic equation.