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The quadratic formula is a powerful tool for finding the solutions of quadratic equations. When an equation is in the form of \( ax^2 + bx + c = 0 \), we can use the quadratic formula to calculate the roots of the equation. The roots are given by:

• \( x = \frac{-b \pm \sqrt{D}}{2a} \)

where \( D \) is the discriminant. This formula works for any quadratic equation, giving us real or complex solutions depending on the value of the discriminant. For our exercise, by using the quadratic formula, we find two real solutions because our discriminant is positive.
This method is reliable and especially useful when the quadratic expression doesn’t factorize easily. It saves time and ensures accuracy in solving complex equations.

The discriminant \( D \) plays a crucial role in understanding the nature of the roots of a quadratic equation. It's calculated as \( D = b^2 - 4ac \), derived from the coefficients of the standard form \( ax^2 + bx + c = 0 \). Here are the implications:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution, or a double root.
• If \( D < 0 \), the solutions are complex or imaginary.

In our case, the discriminant is \( 49 \), which is positive, providing two distinct real roots.
This indicator helps plan the approach to solving the equation, as positive discriminants tell us to expect two different values for \( x \) when we solve using the quadratic formula.

The standard form of a quadratic equation is expressed as \( ax^2 + bx + c = 0 \). This format is crucial as it sets up the equation for operations like factoring, completing the square, and using the quadratic formula. Each term in the equation:

• \( ax^2 \): Quadratic term with \( a \) as the coefficient of \( x^2 \).
• \( bx \): Linear term with \( b \) as the coefficient of \( x \).
• \( c \): Constant term.

By transforming any given quadratic expression into this standard form, we simplify the problem-solving process. In our exercise, we rearranged and adjusted coefficients to achieve \( 12x^2 - 25x + 12 = 0 \), making it easy to apply the discriminant and solve using the quadratic formula.
This rearranging process is key as it prepares the equation for solution using various algebraic methods.