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In solving quadratic equations, the quadratic formula is a powerful tool that functions as a universal solver. With the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), you can find the solutions or roots of any quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. From this formula, you can easily determine the values of \( x \) by using substitution for these constants.
Applying the formula breaks down into a straightforward process:

• Substitute the values of \( a \), \( b \), and \( c \).
• Calculate the discriminant \( b^2 - 4ac \) to understand the nature of the roots.
• Solve for \( x \) by using both the plus and minus signs of the \( \pm \).

This method guarantees a solution, whether the roots are real or complex, making it a crucial method in algebra for resolving quadratic equations.

The discriminant is a key part of the quadratic formula, found under the square root: \( b^2 - 4ac \). Its value tells us important information about the roots:

• If the discriminant is positive, as in our solution with \( \frac{121}{16} \), there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root or a repeated root.
• If the discriminant is negative, the equation has two complex roots.

By calculating \( \left(\frac{5}{4}\right)^2 - 4 \times 1 \times \left(-\frac{3}{2}\right) \), we arrive at \( \frac{121}{16} \), demonstrating a positive discriminant and confirming that our quadratic equation has two distinct real solutions: \( 0.75 \) and \( -2 \). Calculating the discriminant helps to predict the nature of the roots even before calculating them.

Solving quadratic equations often means finding the solutions to equations of the form \( ax^2 + bx + c = 0 \). The solutions, or roots, are the values of \( x \) that satisfy the equation. Here's how you solve it step-by-step:
1. **Rearrange the equation** into the standard form so you can easily identify \( a \), \( b \), and \( c \).2. **Apply the quadratic formula** using the identified coefficients.
For instance, given \( a = 1 \), \( b = \frac{5}{4} \), and \( c = -\frac{3}{2} \), plug them into the formula.3. **Calculate the discriminant** to assess the nature of the roots and proceed to solve for \( x \) using both the plus and minus forms of the quadratic formula's square root component.
For the calculated discriminant of \( 121/16 \), solving leads to the solutions \( x = 0.75 \) and \( x = -2 \). By following these steps, you can confidently solve any quadratic equation and understand the behavior of its solutions.