GED Math: Quadratic Equations
On a multiple-choice question, you can save time with guess-and-check: substitute each answer choice into the equation and see which one makes it true. This is often faster than rewriting, factoring, and solving, and it works well for quadratic equations and expressions. For geometry and number problems, translate the situation into an equation: area = length × width gives a product you can factor, and “the product of two numbers is 56” gives x(x + 1) = 56. Since lengths and counts cannot be negative, reject any negative solution that does not make sense in the situation.
1. What Is a Quadratic Equation?
A quadratic equation has a variable that is squared (and no higher power). In standard form it is written ax2 + bx + c = 0, where a, b, and c are numbers and a is not 0. Unlike a linear equation, which has one solution, a quadratic equation can have two solutions. The most reliable GED method is factoring: set the equation equal to 0, factor the expression into two factors, and find the values of x that make each factor 0. Because a product is 0 only when one of its factors is 0, each of those values is a solution. If an equation is not already equal to 0, move every term to one side first.
Worked Example: Write x2 − x = 20 in standard form and name a, b, and c.
Subtract 20 from both sides: x2 − x − 20 = 0. Here a = 1, b = −1, c = −20.
1. Which of these is a quadratic equation?
Explanation: A quadratic equation contains a squared variable and no higher power. The first has x2.
2. Which is the standard form of x2 − x = 20?
Explanation: Subtract 20 from both sides so that one side equals 0.
3. In 2x2 + 5x − 3 = 0, what are a, b, and c?
Explanation: a is the coefficient of x2, b the coefficient of x, and c the constant, with its sign.
4. What is the greatest number of solutions a quadratic equation can have?
Explanation: Factoring gives two factors, each of which can be 0, so a quadratic equation has at most 2 solutions.
5. Which number is a solution of x2 + 3x + 2 = 0?
Explanation: For x = −1: 1 − 3 + 2 = 0 ✓. For x = 1: 1 + 3 + 2 = 6.
6. Why must you set a quadratic equation equal to 0 before factoring?
Explanation: The zero-product property applies only when a product equals 0.
2. Factoring Out a Common Term
Factoring is the reverse of multiplying: you rewrite an expression as a product of simpler pieces. The first step is always to look for the greatest common factor (GCF), a number, a variable, or both, that divides evenly into every term. Divide each term by the GCF, and write the GCF outside parentheses with the leftovers inside. For the numbers, use the largest number that divides every coefficient. For the variable, use the smallest power that appears in every term. Check by multiplying back with the distributive property.
Worked Example: Factor 15x2 + 9x.
Both terms divide by 3x: 15x2 ÷ 3x = 5x, and 9x ÷ 3x = 3. So the expression is 3x(5x + 3). Check: 3x · 5x + 3x · 3 = 15x2 + 9x ✓.
1. Completely factor 15x2 + 9x
Explanation: The GCF is 3x. Dividing gives 5x and 3. (The first wrong choice is not completely factored: you can still pull out x.)
2. Completely factor 12x2 − 8x
Explanation: The GCF is 4x. 12x2 ÷ 4x = 3x and 8x ÷ 4x = 2, keeping the minus sign.
3. Completely factor 6x2 + 18x
Explanation: The GCF is 6x: 6x2 ÷ 6x = x and 18x ÷ 6x = 3.
4. Completely factor 10x3 − 5x2
Explanation: The GCF is 5x2. 10x3 ÷ 5x2 = 2x and 5x2 ÷ 5x2 = 1.
5. Completely factor x2 − 7x
Explanation: The common factor is x. x2 ÷ x = x and 7x ÷ x = 7.
6. Completely factor 14x2 + 21x
Explanation: The GCF of 14 and 21 is 7, and both terms contain x: 7x(2x + 3).
3. Multiplying Binomials (FOIL) to Check Factoring
To multiply two binomials, multiply First terms, Outer terms, Inner terms, and Last terms, then combine like terms. FOIL is how you check a factoring answer: if multiplying your factors gives back the original expression, your factoring is right. The middle term of the result comes from adding the outer and inner products, which is why the middle coefficient equals the sum of the two numbers in the factors and the last term equals their product: (x + p)(x + q) = x2 + (p + q)x + pq.
Worked Example: Multiply (x + 6)(x − 2).
First: x2. Outer: −2x. Inner: 6x. Last: −12. Combine: x2 + 4x − 12. Notice 6 + (−2) = 4 and 6 × (−2) = −12.
1. Multiply (x+3)(x+4)
Explanation: F: x2; O + I: 4x + 3x = 7x; L: 12.
2. Multiply (x−5)(x−5)
Explanation: F: x2; O + I: −5x − 5x = −10x; L: (−5)(−5) = +25.
3. Multiply (x+6)(x−2)
Explanation: x2 − 2x + 6x − 12 = x2 + 4x − 12.
4. Multiply (2x+1)(x−3)
Explanation: F: 2x2; O: −6x; I: +x; L: −3. Middle: −6x + x = −5x.
5. Multiply (x−7)(x+7)
Explanation: The outer (+7x) and inner (−7x) terms cancel: x2 − 49.
6. Multiply (3x−2)(x+5)
Explanation: 3x2 + 15x − 2x − 10 = 3x2 + 13x − 10.
4. Factoring Trinomials
To factor x2 + bx + c, find two numbers that multiply to c and add to b; those numbers go in the factors: (x + p)(x + q). Use the signs as a guide. If c is positive, the two numbers have the same sign (both positive when b is positive, both negative when b is negative). If c is negative, the numbers have opposite signs, and the larger one takes the sign of b. When the leading coefficient is not 1, as in 2x2 + 5x − 3, use trial and error: try factor pairs of the first and last terms and check with FOIL. Another special pattern is the difference of squares: x2 − 25 = (x − 5)(x + 5).
Worked Example: Factor x2 + 5x + 6.
Which pair of numbers multiplies to 6 and adds to 5? 2 and 3. So the factors are (x + 2)(x + 3). Check: x2 + 3x + 2x + 6 = x2 + 5x + 6 ✓.
1. Factor x2+4x+3
Explanation: Numbers that multiply to 3 and add to 4: 1 and 3.
2. Factor x2+4x−5
Explanation: Multiply to −5 and add to 4: 5 and −1.
3. Factor x2+8x+12
Explanation: Multiply to 12 and add to 8: 2 and 6.
4. Factor x2−x−6
Explanation: Multiply to −6 and add to −1: −3 and 2.
5. Factor x2+5x−14
Explanation: Multiply to −14 and add to 5: 7 and −2.
6. Factor x2−12x+36
Explanation: Multiply to 36 and add to −12: −6 and −6 (a perfect square).
7. Factor x2−6x−7
Explanation: Multiply to −7 and add to −6: −7 and 1.
8. Factor 2x2+5x−3
Explanation: Trial and error: (2x − 1)(x + 3) gives 2x2 + 6x − x − 3 = 2x2 + 5x − 3.
9. Factor 4x2+4x−3
Explanation: (2x − 1)(2x + 3) = 4x2 + 6x − 2x − 3 = 4x2 + 4x − 3.
10. Factor x2−25
Explanation: A difference of squares: the middle terms cancel, so (x − 5)(x + 5).
5. Solving Quadratic Equations by Factoring
To solve a quadratic equation by factoring: (1) move all terms to one side so the equation equals 0; (2) factor the expression; (3) set each factor equal to 0 (the zero-product property: if A × B = 0, then A = 0 or B = 0); (4) solve each small equation; (5) check both answers in the original equation. Never divide both sides by the variable, because you would lose a solution (for example, dividing 3x2 = 12x by 3x hides the solution 0). The solutions are exactly where the graph of the quadratic crosses the x-axis.
Worked Example: Solve x2 − x = 20.
Set equal to 0: x2 − x − 20 = 0. Factor: (x − 5)(x + 4) = 0. Set each factor to 0: x − 5 = 0 gives x = 5, and x + 4 = 0 gives x = −4. Solutions: −4 and 5. Check: 25 − 5 = 20 ✓ and 16 + 4 = 20 ✓.
1. What are the two solutions of x2−x=20?
Explanation: Move 20 over: x2 − x − 20 = 0, which factors as (x − 5)(x + 4) = 0. So x = 5 or x = −4.
2. Solve x2+4x+3=0
Explanation: Factor: (x + 1)(x + 3) = 0. Set each to 0: x = −1 or x = −3.
3. Solve x2+5x=14
Explanation: Subtract 14: x2 + 5x − 14 = 0. Factor: (x + 7)(x − 2) = 0. So x = −7 or x = 2.
4. Solve x2=5x+24
Explanation: Move terms: x2 − 5x − 24 = 0. Factor: (x − 8)(x + 3) = 0. So x = 8 or x = −3.
5. Solve x2+3x=4
Explanation: Subtract 4: x2 + 3x − 4 = 0. Factor: (x + 4)(x − 1) = 0. So x = −4 or x = 1.
6. Solve 3x2=12x
Explanation: Subtract 12x: 3x2 − 12x = 0. Factor out 3x: 3x(x − 4) = 0. So x = 0 or x = 4. Do not divide by x or you lose x = 0.
7. Solve x2−49=0
Explanation: Difference of squares: (x − 7)(x + 7) = 0, so x = 7 or x = −7.
8. What is the only positive solution of 2x2 − 7x − 30 = 0?
Explanation: Factor: (2x + 5)(x − 6) = 0. So x = −5/2 or x = 6. The only positive one is 6.
9. For which equation is x = −4 a solution?
Explanation: Substitute x = −4: 16 − 8 − 8 = 0 ✓. The others give 24, 112, and 9.
6. Guess-and-Check and Quadratic Word Problems
On a multiple-choice question, you can save time with guess-and-check: substitute each answer choice into the equation and see which one makes it true. This is often faster than rewriting, factoring, and solving, and it works well for quadratic equations and expressions. For geometry and number problems, translate the situation into an equation: area = length × width gives a product you can factor, and “the product of two numbers is 56” gives x(x + 1) = 56. Since lengths and counts cannot be negative, reject any negative solution that does not make sense in the situation.
Worked Example: Which is a solution of 2x2 − 12 = 2x? (A) 4 (B) 3 (C) 0 (D) −1.
Try each: for 4, 2(16) − 12 = 20 but 2(4) = 8, so no. For 3, 2(9) − 12 = 6 and 2(3) = 6, which matches. The answer is (B) 3.
1. Which is a solution of 2x2 − 12 = 2x?
Explanation: For x = 3: 2(9) − 12 = 6 and 2(3) = 6 ✓. For x = 4: 20 vs 8; x = 0: −12 vs 0; x = −1: −10 vs −2.
2. The area of a rectangle is 2x2 − 27x + 70. Its length is 2x − 7. Which expression is the width?
Explanation: Factor the area: 2x2 − 27x + 70 = (2x − 7)(x − 10). The width is x − 10.
3. The area of a rectangle is x2 + 9x + 20 and its length is x + 5. Which expression is the width?
Explanation: x2 + 9x + 20 = (x + 5)(x + 4), so the width is x + 4.
4. A garden’s length is 3 feet more than its width w. Its area is 40 square feet. What is the width?
Explanation: w(w + 3) = 40 gives w2 + 3w − 40 = 0, which factors as (w + 8)(w − 5) = 0. Width cannot be negative, so w = 5.
5. The product of two consecutive positive integers is 56. What is the smaller integer?
Explanation: x(x + 1) = 56, so x2 + x − 56 = 0, which factors as (x + 8)(x − 7) = 0. The positive solution is 7 (7 × 8 = 56).
6. A positive number plus its square equals 30. What is the number?
Explanation: x2 + x = 30 gives (x + 6)(x − 5) = 0. The positive solution is 5: 25 + 5 = 30.
Common Mistakes to Avoid
- Not setting the equation equal to 0 before factoring.
- Dividing both sides by x, which loses the solution x = 0.
- Choosing a pair of numbers that multiply to c but do not add to b. Check both conditions.
- Getting the signs wrong in the factors. Check by multiplying back with FOIL.
- Writing the solutions with the wrong signs: the factor (x + 4) gives x = −4.
- Leaving out one of the two solutions.
- Not factoring out the greatest common factor first, so the expression is not completely factored.
- Keeping a negative answer in a real-world problem where lengths and counts must be positive.
Frequently Asked Questions
What is a quadratic equation?
An equation with a squared variable that can be written as ax2 + bx + c = 0. It can have up to two solutions.
How do I solve a quadratic equation by factoring?
Set it equal to 0, factor, set each factor equal to 0, and solve each small equation.
How do I factor x2 + bx + c?
Find two numbers that multiply to c and add to b, and write them in (x + p)(x + q).
What is the zero-product property?
If two factors multiply to 0, at least one of them must equal 0.
When should I use guess-and-check?
On multiple-choice questions, substitute each answer choice into the original equation. It is often faster than solving.
What does the graph of a quadratic show about its solutions?
The solutions are the x-values where the graph crosses the x-axis.
Where can I practice more problems like these?
The GED Math practice in the GED Math QBank gives you more practice on this topic, and you can browse every other topic in the full GED Math QBank.