GED Math Quadratic Equations Summary

1. Take the square root of both sides.
2. Both 7 × 7 and (−7) × (−7) equal 49.
3. So x = 7 or x = −7.

Check: 7² = 49 and (−7)² = 49.

Wrong choices

• x = 7 only: x = 7 only misses the negative answer; (−7)² is also 49.
• x = 24.5 or x = −24.5: x = 24.5 or x = −24.5 divides 49 by 2 instead of taking the square root.
• x = 2,401: x = 2,401 squares 49 instead of taking its square root.

1. Set each factor equal to 0.
2. x − 3 = 0 gives x = 3.
3. x + 5 = 0 gives x = −5.

Check: (3 − 3)(3 + 5) = 0 × 8 = 0, and (−5 − 3)(−5 + 5) = −8 × 0 = 0.

Wrong choices

• x = −3 or x = 5: x = −3 or x = 5 takes the numbers with the signs shown in the factors; the signs must be reversed.
• x = 3 or x = 5: x = 3 or x = 5 gets the sign of the second answer wrong; 5 + 5 is not 0.
• x = −3 or x = −5: x = −3 or x = −5 gets the sign of the first answer wrong; −3 − 3 is not 0.

1. Find two numbers that multiply to 12 and add to −7: −3 and −4.
2. Factor: (x − 3)(x − 4) = 0.
3. x = 3 or x = 4.

Check: 3² − 7(3) + 12 = 9 − 21 + 12 = 0, and 4² − 7(4) + 12 = 16 − 28 + 12 = 0.

Wrong choices

• x = −3 or x = −4: x = −3 or x = −4 uses the numbers from the factors without reversing the signs; (−3)² − 7(−3) + 12 = 42.
• x = 2 or x = 6: x = 2 or x = 6 multiply to 12 but add to 8, not 7; 2² − 14 + 12 = 2.
• x = 1 or x = 12: x = 1 or x = 12 multiply to 12 but add to 13; 1 − 7 + 12 = 6.

1. Substitute: y = (−2)² − 4(−2) + 1.
2. (−2)² = 4.
3. −4 × (−2) = +8.
4. y = 4 + 8 + 1 = 13.

Check: Put x = −2 into the calculator with parentheses: (−2)² − 4 × (−2) + 1 = 13.

Wrong choices

• −3: −3 treats −4(−2) as −8; a negative times a negative is positive.
• 5: 5 treats (−2)² as −4; a negative number squared is positive.
• −11: −11 makes both sign mistakes: −4 − 8 + 1.

1. Divide both sides by 3: x² = 25.
2. Take the square root: x = 5 or x = −5.

Check: 3 × 5² = 3 × 25 = 75, and 3 × (−5)² = 75.

Wrong choices

• x = 25 or x = −25: x = 25 or x = −25 stops at x² = 25 and forgets the square root.
• x = 5 only: x = 5 only misses the negative answer.
• x = 8.66 or x = −8.66: x = 8.66 or x = −8.66 takes the square root of 75 before dividing by 3.

1. a = 2, b = 3, c = −2.
2. b² − 4ac = 9 − 4(2)(−2) = 9 + 16 = 25. √25 = 5.
3. x = (−3 + 5) / 4 = 2/4 = 1/2.
4. x = (−3 − 5) / 4 = −8/4 = −2.

Check: Factor: 2x² + 3x − 2 = (2x − 1)(x + 2), which is 0 when x = 1/2 or x = −2.

Wrong choices

• x = −1/2 or x = 2: x = −1/2 or x = 2 uses +b instead of −b at the start of the formula.
• x = 1 or x = −4: x = 1 or x = −4 divides by 2 instead of 2a = 4.
• x = 1/4 or x = −1: x = 1/4 or x = −1 divides by 8 (4a) instead of 2a = 4.

1. Let w = width. Length = w + 4.
2. Area: w(w + 4) = 96, so w² + 4w − 96 = 0.
3. Factor: (w + 12)(w − 8) = 0.
4. w = 8 or w = −12. A width cannot be negative, so w = 8 feet.

Check: 8 × 12 = 96 square feet, and 12 is 4 more than 8.

Wrong choices

• 12 feet: 12 feet is the length, not the width.
• 24 feet: 24 feet divides the area by 4; the 4 is the difference between length and width, not a side.
• 10 feet: A 10-foot width gives a 14-foot length and an area of 140 square feet.

1. Subtract 16 from both sides: x² + 6x − 16 = 0.
2. Find two numbers that multiply to −16 and add to 6: 8 and −2.
3. Factor: (x + 8)(x − 2) = 0.
4. x = −8 or x = 2.

Check: 2² + 6(2) = 4 + 12 = 16, and (−8)² + 6(−8) = 64 − 48 = 16.

Wrong choices

• x = −2 or x = 8: x = −2 or x = 8 reverses both signs; (−2)² + 6(−2) = −8, not 16.
• x = 0 or x = −6: x = 0 or x = −6 solves x² + 6x = 0, forgetting the 16.
• x = 16 or x = 10: x = 16 or x = 10 sets x = 16 and x + 6 = 16, which only works when the right side is 0.

1. On the ground, h = 0: −16t² + 64t = 0.
2. Factor: −16t(t − 4) = 0.
3. t = 0 (when it is kicked) or t = 4.
4. It lands after 4 seconds.

Check: h = −16(4)² + 64(4) = −256 + 256 = 0.

Wrong choices

• 2 seconds: At 2 seconds the ball is at its highest point (64 feet), not on the ground.
• 64 seconds: 64 is the greatest height in feet, not a time.
• 16 seconds: At 16 seconds the formula gives a large negative height, so the ball landed well before that.

1. Consecutive whole numbers differ by 1, so the numbers are n and n + 1.
2. "Product" means multiply: n(n + 1).
3. Equation: n(n + 1) = 132.

Check: n = 11 works, because 11 × 12 = 132.

Wrong choices

• n + (n + 1) = 132: n + (n + 1) = 132 adds the numbers; the problem says product.
• n² = 132: n² = 132 multiplies the smaller number by itself, not by the next number.
• n(n + 2) = 132: n(n + 2) = 132 is for consecutive even or odd numbers, which differ by 2.