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The quadratic formula is a powerful tool designed specifically to solve any quadratic equation, which takes the general form \( ax^2 + bx + c = 0 \). Whether you're dealing with a simple or complex quadratic, this formula allows you to find the exact solutions, or roots, for the equation. The formula is expressed as:

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

Here, \( a \), \( b \), and \( c \) represent coefficients from the quadratic equation itself. Once you have these values identified, you can plug them into the formula to determine which values of \( x \) make the equation true. You'll often see the "\( \pm \)" symbol, which implies that there will typically be two solutions: one that adds the square root and one that subtracts it. This makes the quadratic formula versatile in finding the solutions for a variety of quadratic equations.

The discriminant is an integral part of the quadratic formula. It is the component under the square root sign, expressed as \( b^2 - 4ac \). Calculating the discriminant is essential because it indicates the nature of the roots:

• If the discriminant is positive, you will have two distinct real roots.
• If it equals zero, there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, the roots are not real numbers but complex or imaginary roots.

For example, when calculating the discriminant for the equations provided:

• Equation: \( a^2 + a - 7 = 0 \) gives \( b^2 - 4ac = 29 \)
• Equation: \( a^2 - a - 7 = 0 \) also gives \( b^2 - 4ac = 29 \)

Both results are positive, confirming that each equation possesses two distinct real roots.

The roots of a quadratic equation are the solutions for the variable that make the equation equal to zero. Once the discriminant is calculated and found to be non-negative, finding the roots becomes straightforward. With a positive discriminant, apply the quadratic formula:

• For \( a^2 + a - 7 = 0 \), the roots are: \( a = \frac{-1 + \sqrt{29}}{2} \) and \( a = \frac{-1 - \sqrt{29}}{2} \)
• For \( a^2 - a - 7 = 0 \), the roots are: \( a = \frac{1 + \sqrt{29}}{2} \) and \( a = \frac{1 - \sqrt{29}}{2} \)

Each equation yields two roots due to the presence of the "\( \pm \)" in the quadratic formula which accounts for the two possible values of \( x \). These roots are visually the points where the parabola defined by the quadratic equation crosses the x-axis.