Quadratics

• Factor quadratic expressions of the form ax² + bx + c
• Solve quadratic equations by factoring and setting each factor to zero
• Apply the quadratic formula x = (−b ± √(b² − 4ac)) / 2a, and use the discriminant b² − 4ac to count the real solutions
• Solve word problems that produce quadratic equations
• Finish on one ACT-hard question and say what made it hard

A quadratic equation has the form ax² + bx + c = 0, with a ≠ 0. To factor, find two numbers that multiply to ac and add to b, then rewrite and group. The real solutions (roots) are the x-values where the parabola meets the x-axis: it crosses at a simple root and touches at a repeated root. The quadratic formula x = (−b ± √(b² − 4ac)) / 2a works for any quadratic. The discriminant b² − 4ac tells you the number of solutions: positive → two real solutions, zero → one repeated solution, negative → no real solutions. The ACT tests factoring and quadratics on several questions per test, making it one of the highest-yield math topics to study. For 'which expression is equivalent' questions, pick a number: substitute a value such as x = 2 into the original expression and into each choice; equivalent expressions agree for every value of x, so any choice that disagrees is out (use a second value if two choices agree). The hardest questions, at the end of the section: questions solved by structure rather than by grinding. For ax² + bx + c = 0 the solutions add to −b/a and multiply to c/a, so the sum of the solutions of 2x² − 6x − 7 = 0 is 3 without solving; and recognizing a perfect square such as x² − 10x + 25 = (x − 5)² shows at once that there is exactly one solution. A squared binomial always expands as (x + k)² = x² + 2kx + k², so if (x + k)² = x² + 58x + k², match the middle terms: 2k = 58 and k = 29. Word problems that give a quadratic: a rectangle's length is 3 more than its width and its area is 40, so w(w + 3) = 40, w² + 3w − 40 = 0, (w + 8)(w − 5) = 0, and w = 5 (reject −8: a length cannot be negative), so the rectangle is 5 by 8. Always check which root the situation allows.

• If the leading coefficient is 1, you only need two numbers that multiply to c and add to b.
• Memorize the quadratic formula. You will need it when factoring is not straightforward.
• Always check your solutions by substituting back into the original equation.