ACT Math: Completing the Square Unlocks Vertex, Roots, and Graphs

Step 1: Start with ax²+bx+c. Step 2: Factor 'a' from the first two terms: a(x²+(b/a)x)+c. Step 3: Take half the coefficient of x and square it: (b/2a)². Step 4: Add and subtract this inside the parentheses: a(x²+(b/a)x+(b/2a)²-(b/2a)²)+c. Step 5: Simplify to a(x+b/2a)²+(c-a(b/2a)²). This gives you vertex form immediately. The vertex is (-b/2a, c-a(b/2a)²), which you can read directly from the final form.

Example: y=2x²+8x+3. Factor 2: y=2(x²+4x)+3. Half of 4 is 2; 2²=4. Add/subtract: y=2(x²+4x+4-4)+3. Simplify: y=2((x+2)²-4)+3=2(x+2)²-8+3=2(x+2)²-5. Vertex: (-2,-5).

Error 1: Forgetting to factor 'a' before completing the square. If a≠1, you must factor it first, or your algebra breaks. Error 2: Squaring the wrong number. Half the coefficient of x is (b/2a), not b/2. Error 3: Losing track of signs. When adding (b/2a)² inside, subtract it afterward: +(...)-(...). Error 4: Not simplifying the constant term fully. After expanding, always combine like terms. Double-check your algebra by expanding your vertex form back to standard form; it should match the original.

Always verify: If y=2(x+2)²-5, expand: 2(x²+4x+4)-5=2x²+8x+8-5=2x²+8x+3. Matches original. ✓

Problem 1: y=x²+6x+5. Step-by-step: y=(x²+6x)+5. Half of 6 is 3; 3²=9. y=(x²+6x+9-9)+5=(x+3)²-9+5=(x+3)²-4. Vertex: (-3,-4). Problem 2: y=3x²+12x+1. Factor 3: y=3(x²+4x)+1. Half of 4 is 2; 2²=4. y=3(x²+4x+4-4)+1=3((x+2)²-4)+1=3(x+2)²-12+1=3(x+2)²-11. Vertex: (-2,-11). Problem 3: y=-x²+4x+7. Factor -1: y=-(x²-4x)+7. Half of -4 is -2; (-2)²=4. y=-(x²-4x+4-4)+7=-((x-2)²-4)+7=-(x-2)²+4+7=-(x-2)²+11. Vertex: (2,11). Do these three daily for one week until your speed and accuracy improve.

Time yourself. You should complete each problem in under 90 seconds.

Completing the square appears in 2-3 questions per ACT Math section and is fundamental to understanding parabolas. Once you own this skill, you can find the vertex without the formula, derive the axis of symmetry, and transform any quadratic into the most useful form. Many hard quadratic questions require completing the square; mastering it bumps you from guessing to confident problem-solving.

Invest one week in this. By test day, completing the square should feel like automatic algebra, freeing your mind for harder strategy.