b. The Quadratic Formula calculates two possible answers by using the " \( \pm \) " symbol. This symbol (read as "plus or minus") is shorthand notation that tells you to calculate the formula twice: once with addition and once with subtraction in the numerator. Therefore, every Quadratic Formula problem is really two different problems unless the value of \( \sqrt{b^{2}-4 a_{c}} \) is 0 . \[ x=\frac{-b+\sqrt{b^{2}-4 a c}}{2 a} \text { or } x=\frac{-b-\sqrt{b^{2}-4 a c}}{2 a} \] Carefully substitute \( a, b \), and \( c \) from \( x^{2}-3 x-7=0 \) into the Quadratic Formula. Evaluate each expression (once using addition and once using subtraction) to solve for \( x \). Do these solutions match those from problem \( 5-95 \) ?

To solve the quadratic equation \( x^2 - 3x - 7 = 0 \), we identify the coefficients as \( a = 1 \), \( b = -3 \), and \( c = -7 \). Plug these values into the quadratic formula. First, calculate the discriminant: \( b^2 - 4ac = (-3)^2 - 4(1)(-7) = 9 + 28 = 37 \). Now, you can use this in the formula: \[ x = \frac{-(-3) \pm \sqrt{37}}{2(1)} = \frac{3 \pm \sqrt{37}}{2} \] Thus, the two solutions are \( x = \frac{3 + \sqrt{37}}{2} \) and \( x = \frac{3 - \sqrt{37}}{2} \). After calculating the values from problem \( 5-95 \), check if they are equal to these results! If so, congrats, you’ve verified your solution! Now, let's spice this up a bit! Did you know that quadratic equations can also be visualized as parabolas? Each solution you find via the quadratic formula corresponds to the points where the parabola intersects the x-axis. So, if you plotted \( x^2 - 3x - 7 \), you’d see a U-shaped curve touching the x-axis at those two solutions. Additionally, mastering the quadratic formula provides a solid foundation not just for solving equations but also for tackling real-world problems like projectile motion, where the height of an object can be modeled by a quadratic equation. By understanding how and when to apply the quadratic formula, you enable yourself to solve everything from designing roller coasters to predicting the trajectory of rockets! So get ready for some math magic in real life!