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The quadratic formula is a handy tool for solving quadratic equations, which are equations in the form of \( ax^2 + bx + c = 0 \). This formula provides the roots, or solutions, for any quadratic equation:

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

Here, \( a \), \( b \), and \( c \) represent numerical coefficients in the equation. It's essential to substitute these values accurately into the formula to find the correct solutions. The quadratic formula is beneficial because it works for all forms of quadratic equations, even when the roots aren’t simple integers.
To solve the example equation \( m^2 - 7m + 3 = 0 \), we identify \( a = 1 \), \( b = -7 \), and \( c = 3 \). Substituting these values into the quadratic formula will help us find the roots of this equation.

The discriminant is a key part of the quadratic formula, found under the square root symbol: \( b^2 - 4ac \). This part of the formula is crucial because it tells us the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root, often referred to as a repeated root.
• If the discriminant is negative, there are no real roots - instead, the roots are complex numbers.

In our exercise, the discriminant calculated as \( 37 \) indicates there are two real and distinct roots. Knowing about the discriminant helps predict the type of solutions you will encounter, allowing you to anticipate whether further simplifications or approximations might be necessary.

Real roots are solutions to the quadratic equation that are real numbers, meaning they can be plotted on a number line. After determining from the discriminant that our equation has two real roots, we substitute the values into the quadratic formula to solve for them.

• In the current problem, we resolve to find \( m = \frac{7 \pm \sqrt{37}}{2} \).
• Calculating \( \sqrt{37} \) gives approximately \( 6.08276 \).

This leads to the two roots:

• \( m_1 = \frac{7 + 6.08276}{2} \approx 6.54 \)
• \( m_2 = \frac{7 - 6.08276}{2} \approx 0.46 \)

These real roots signify the values of \( m \) that satisfy the original equation \( m^2 - 7m + 3 = 0 \). Verifying these roots by plugging them back into the original equation also confirms their correctness. Understanding how real roots work helps in finding meaningful numerical solutions that correspond to real-world problems and scenarios.