At the end of this lesson, students will be able to:
Solve quadratic equations using the quadratic formula.
Identify and choose methods for solving quadratic equations.
Solve real-world problems using functions by completing the square.
Terms introduced in this lesson:
roots, solutions to a quadratic equation
Teaching Strategies and Tips
Emphasize that the quadratic formula comes from completing the square of a general quadratic equation.
Point out that half the coefficient of x squared can be found by multiplying by 1/2:
Remind students to find common denominators before simplifying the right side of the equation.
Have students rewrite each quadratic equation in standard form first. See Example 3.
Use Example 4 to show students how one goes about choosing a solving method.
Teachers are encouraged to use quadratic equations with non-integer coefficients—decimals and fractions—as additional practice problems.
General Tip: Some students may use the values of a, b, and c based upon the order in which the terms were given. Students will assume that a is the first coefficient, b the second, and c the last.
Remind students rewrite each quadratic equation in standard form first. See Example 3.
General Tip: Have students simplify one step at a time when using the quadratic formula:
Simplify under the radical.
Determine the square root.
Simplify the numerator.
Simplify the denominator.
General Tip: Some common mistakes associated with the quadratic formula are:
Not using the minus sign that goes with a coefficient.
Example: 3x2+7x−4=0. Use a=3,b=7, and c=−4.(c≠4)
Losing the minus sign on −b.
Example: 3x2−7x+4=0. Use a=3,b=−7, and c=4.
x=−(−7)±(−7)2−4(3)(4)−−−−−−−−−−−−√2(3) and not x=−7±(−7)2−4(3)(4)−−−−−−−−−−−−√2(3)
Canceling a factor of the denominator with −b only. Remind students that in order to cancel a factor from the denominator, it must be a common factor in every term in the numerator. Unless that factor comes out of the radical, then the factor in the denominator cannot be canceled.
Example: For a=3, b=6, and c=−1, x=−6±62−4(3)(−1)−−−−−−−−−−−√2(3)=−6±48−−√6⇏x=−6±48−−√6.