21) \( b^{2}-17 b+70 \)

To factor the quadratic \( b^{2}-17b+70 \), we look for two numbers that multiply to 70 and add up to -17. Those numbers are -7 and -10. Therefore, we can express the quadratic as \( (b-7)(b-10) \). Now, set each factor to zero to find the solutions: \( b - 7 = 0 \) gives \( b = 7 \) and \( b - 10 = 0 \) gives \( b = 10 \). Thus, the solutions to the equation are \( b = 7 \) and \( b = 10 \). In the real world, factoring quadratics like \( b^{2}-17b+70 \) can be especially useful in various fields including physics, engineering, and finance. For example, if you were to model the trajectory of a projectile, the quadratic equations can represent height, while the roots could indicate when the projectile hits the ground. Understanding these roots can help in planning and decision-making. One common mistake when factoring quadratics is overlooking the signs of the numbers. Always remember to consider both the signs and values that need to match both the product and sum conditions to successfully factor the expression. It can be helpful to write out the pairs of factors and test them to ensure they fulfill the quadratic's requirements!