4.1 Factoring & Solving Quadratics
Key Takeaways
- The zero-product property turns a factored quadratic into two linear equations: if A times B equals 0, then A equals 0 or B equals 0.
- The quadratic formula x = (-b plus or minus the square root of (b^2 - 4ac)) / (2a) solves any equation written as ax^2 + bx + c = 0.
- The discriminant b^2 - 4ac decides the roots: positive gives two real roots, zero gives one real (double) root, negative gives no real roots.
- Always factor a common GCF first; difference of squares a^2 - b^2 factors as (a - b)(a + b).
- Completing the square rewrites ax^2 + bx + c so the variable appears once, exposing exact and irrational roots.
Why Quadratics Carry the Exam
The Algebra strand is 48%-61% of Regents credits and Functions is 24%-32%, and quadratics sit in both. You will factor expressions, solve quadratic equations several ways, and read parabolas. Mastering this one topic protects a large block of points across Parts I-IV.
A quadratic equation has the standard form ax^2 + bx + c = 0 with a not equal to 0. The graph of y = ax^2 + bx + c is a parabola. The x-values that make the expression equal 0 are the roots (also called zeros or solutions).
Factoring Toolkit
Factoring rewrites a sum as a product so you can apply the zero-product property. Work through these in order.
- GCF first. Pull out the greatest common factor: 2x^2 + 10x = 2x(x + 5).
- Trinomials, leading coefficient 1. For x^2 + bx + c, find two numbers that multiply to c and add to b. For x^2 + 7x + 10, the pair 2 and 5 works: (x + 2)(x + 5).
- Difference of squares. a^2 - b^2 = (a - b)(a + b). So x^2 - 9 = (x - 3)(x + 3).
- Leading coefficient not 1. Use the AC method below.
The AC Method (a not equal to 1)
To factor 2x^2 + 7x + 3: multiply a times c = 2 times 3 = 6. Find two numbers that multiply to 6 and add to b = 7: that is 6 and 1. Split the middle term:
2x^2 + 6x + 1x + 3, then group: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
Check by expanding: 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3. Correct.
The Zero-Product Property
This is the engine behind solving by factoring. If a product equals zero, at least one factor equals zero. Symbolically, if A times B = 0, then A = 0 or B = 0.
Solve x^2 - 5x + 6 = 0. Factor to (x - 2)(x - 3) = 0. Set each factor to zero: x - 2 = 0 gives x = 2; x - 3 = 0 gives x = 3. The solution set is {2, 3}.
A common Regents trap: x^2 = 9x. Do NOT divide by x (you lose a root). Move everything to one side: x^2 - 9x = 0, factor x(x - 9) = 0, so x = 0 or x = 9.
What is the complete solution set of 2x^2 - 3x - 2 = 0?
Completing the Square
When a quadratic does not factor nicely, completing the square rewrites it so the variable appears once. The key value to add is (b/2)^2.
Solve x^2 + 10x = -9 (a = 1). Take half of b = 10, giving 5; square it to get 25. Add 25 to both sides:
x^2 + 10x + 25 = -9 + 25, so (x + 5)^2 = 16. Take the square root: x + 5 = plus or minus 4. Thus x = -1 or x = -9.
The blank that completes x^2 + 10x + ___ is 25.
The Quadratic Formula and Discriminant
The quadratic formula solves any ax^2 + bx + c = 0 and is on the Regents reference sheet:
x = (-b plus or minus the square root of (b^2 - 4ac)) / (2a)
The expression under the radical, b^2 - 4ac, is the discriminant. It tells you how many real roots exist before you finish solving.
Discriminant Rules
| Discriminant b^2 - 4ac | Number of real roots | Root type |
|---|---|---|
| Greater than 0 | Two | Two distinct real roots |
| Equal to 0 | One | One real (double) root |
| Less than 0 | None | No real roots (two complex) |
Worked example: x^2 + 4x + 7 = 0. Here a = 1, b = 4, c = 7, so b^2 - 4ac = 16 - 28 = -12. Because -12 is less than 0, the equation has no real solutions, and its parabola never crosses the x-axis.
Worked Formula Example
Solve x^2 - 6x + 4 = 0 with the formula. a = 1, b = -6, c = 4.
Discriminant: (-6)^2 - 4(1)(4) = 36 - 16 = 20 (positive, so two real roots).
x = (6 plus or minus the square root of 20) / 2. Since the square root of 20 = 2 times the square root of 5, this simplifies to x = (6 plus or minus 2 times the square root of 5) / 2 = 3 plus or minus the square root of 5.
Solving by Graphing
On the Regents you may use the graphing calculator. The real roots are the x-intercepts of y = ax^2 + bx + c. Graph the function and read where it crosses the x-axis. If the parabola does not cross the x-axis, the equation has no real roots, matching a negative discriminant.
For a system such as y = x + 2 and y = x^2 - 4, the solutions are the intersection points. Set x + 2 = x^2 - 4, giving x^2 - x - 6 = 0, factor (x - 3)(x + 2) = 0, so x = 3 (point (3, 5)) and x = -2 (point (-2, 0)). Graphing confirms two crossings.
Choosing a method: factor when the trinomial is simple, complete the square to find exact irrational roots or a vertex, use the formula when factoring fails, and graph to verify or to read a system's solutions quickly.
How many real solutions does 3x^2 - 4x + 5 = 0 have, based on the discriminant?