7.3 Systems, Quadratics, and Polynomials
Key Takeaways
- GED system questions ask for the point or values that satisfy two linear equations at the same time.
- Substitution is efficient when one equation already has x or y isolated; elimination is efficient when coefficients line up.
- Quadratic equations on the GED have rational coefficients and real solutions, and can be solved by factoring, inspection, or the quadratic formula.
- Polynomial work includes combining like terms, multiplying binomials, factoring, and evaluating expressions by substitution.
- The fastest GED strategy is to choose the method that fits the form instead of using the same method every time.
GED Focus
The official GED algebra targets include solving systems of two linear equations, solving quadratic equations with real solutions, and computing with polynomials. These topics look different, but they share one test-day goal: find a value, expression, or ordered pair that makes the math true.
Systems: One Solution For Two Conditions
A system of equations contains two equations that must both be true. On a graph, the solution is the intersection point. Algebraically, the solution is the x-value and y-value that satisfy both equations.
| Method | Best When | Quick Signal |
|---|---|---|
| Graphing | The graph is already provided | Lines cross at a clear point |
| Substitution | One variable is isolated | y = 2x + 1 |
| Elimination | Coefficients match or oppose | 3x + 2y and 3x - 5y |
Worked Example: Elimination
Solve the system:
2x + y = 17 3x - y = 8
Add the equations so y and -y cancel: 5x = 25, so x = 5. Substitute into the first equation: 2(5) + y = 17, so y = 7. The solution is (5, 7). Check in the second equation: 3(5) - 7 = 8, so it works.
Polynomials: Treat Like Terms Carefully
A polynomial is an expression made of terms such as 4x^2, -3x, and 9. GED polynomial items may ask you to add, subtract, multiply binomials, divide factorable polynomials, evaluate by substitution, or factor.
Add like terms only when the variable part matches exactly. For example, 6x^2 + 3x - 2x^2 + 8x = 4x^2 + 11x. Do not combine x^2 and x terms.
To multiply binomials, multiply every term in the first factor by every term in the second factor. Example: (x + 4)(x + 3) = x^2 + 3x + 4x + 12 = x^2 + 7x + 12.
Quadratics: Set Equal To Zero
A quadratic equation has an x^2 term and can often be written as ax^2 + bx + c = 0. Many GED quadratics are factorable.
Solve x^2 + 7x + 12 = 0.
Find two numbers that multiply to 12 and add to 7: 3 and 4. Factor: (x + 3)(x + 4) = 0. Set each factor equal to zero: x + 3 = 0 or x + 4 = 0. The solutions are x = -3 and x = -4.
If factoring is not clear, use the quadratic formula when the answer choices are numeric and the equation is in standard form. You do not need to memorize every formula sheet item blindly; you do need to know which numbers are a, b, and c.
Interpreting Answers
Systems usually give an ordered pair, so label both coordinates in context. Quadratics may give two solutions, but a word problem may accept only one. If a rectangle width comes out as -4 or 7, the negative value is rejected because a physical width cannot be negative.
Test-Day Method Choice
- Use substitution when a variable is already alone.
- Use elimination when adding or subtracting equations cancels a variable.
- For quadratics, first check whether the equation is set equal to zero.
- For polynomials, underline like terms before combining.
- For word problems, interpret the solution. Negative time, negative length, or half a person may be algebraically possible but not realistic.
Solve the system: x + y = 14 and x - y = 2.
Factor x^2 + 9x + 20.