Common Core Standards: Math
4. Solve quadratic equations in one variable.
Students should know how to solve equations involving terms with one variable to the second degree. These equations may be written in any form, the most common being the standard form of a quadratic equation, ax2 + bx + c = 0. Students should know and apply the three main ways to solve a quadratic equation: stop, drop, and roll.
Oh, wait. Scratch that. We meant factor, complete the square, and the quadratic formula.
Your students should already know what factoring is, and that it's possible with simple quadratic equations like x2 + x – 12 = 0. When factoring, students should look for two numbers that add to the coefficient b (in this case, 1) and multiply to get the constant c (in this case, -12). Easier said than done, unless you've got a mouthful of peanut butter.
For the equation x2 + x – 12, the two numbers that work are -3 and 4. They add to get 1 and multiply to get -12, so we can factor x2 + x – 12 into (x – 3)(x + 4). Now it's way easier to solve when we set the equation to equal 0. Since x – 3 = 0 and x + 4 = 0, our answers are x = 3 and x = -4. Note that this method only works when a is 1.
- Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
- Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.