7.3 Systems, Quadratics, and Polynomials
Key Takeaways
- A GED system asks for the single ordered pair (x, y) that satisfies two linear equations at the same time; graphically it is the intersection point.
- Use substitution when one variable is already isolated, and elimination when coefficients line up to cancel by adding or subtracting.
- GED quadratics have rational coefficients and real solutions; solve by factoring, by inspection, or with the quadratic formula given on the formula sheet.
- Polynomial work covers combining like terms, multiplying binomials with FOIL, factoring trinomials, and evaluating by substitution.
- Match the method to the form rather than forcing one technique on every problem; checking a context answer can reject an impossible negative solution.
GED Focus: Find the Values That Make It True
The official GED algebra targets here include solving systems of two linear equations, solving quadratic equations with real solutions, and computing with polynomials. The topics look different but share one goal: find a value, expression, or ordered pair that makes every statement true.
The GED gives you an on-screen formula sheet that includes the quadratic formula, x = (-b plus or minus the square root of (b^2 - 4ac)) / (2a). You do not have to memorize it, but you must be able to identify a, b, and c from a quadratic written in standard form ax^2 + bx + c = 0.
The formula sheet also lists area, volume, and the Pythagorean relationship, but it does NOT teach you when or how to apply them. Knowing which sheet item fits a given algebra problem is the skill the GED actually measures.
Systems: One Solution for Two Conditions
A system of equations is two equations that must both hold. On a graph, the solution is the point where the lines cross. Algebraically, it is the (x, y) pair satisfying both.
| Method | Best when | Quick signal |
|---|---|---|
| Graphing | A graph is already provided | Lines cross at a clear lattice point |
| Substitution | One variable is isolated | y = 2x + 1 |
| Elimination | Coefficients match or oppose | 3x + 2y and 3x - 5y |
If both equations reduce to the same line, there are infinitely many solutions; if they give parallel lines (same slope, different intercept), there is no solution. The GED occasionally tests recognizing these special cases by showing two equations and asking how many solutions exist, so compare slopes: equal slopes with different intercepts means zero solutions, while different slopes always means exactly one solution.
Worked Example: Elimination
Solve the system:
2x + y = 17 3x - y = 8
The y terms are +y and -y, so add the two equations to cancel y: 5x = 25, giving x = 5. Substitute into the first equation: 2(5) + y = 17, so y = 7. The solution is the ordered pair (5, 7). Always verify in the OTHER equation: 3(5) - 7 = 15 - 7 = 8, which matches. When coefficients are not opposites, multiply one equation through by a constant first; for 2x + 3y = 12 and x + y = 5, multiply the second by 2 to line up the x terms.
Polynomials: Combine Like Terms Carefully
A polynomial is built from terms such as 4x^2, -3x, and 9. GED items may ask you to add, subtract, multiply binomials, or evaluate by substitution. Combine like terms only when the variable AND its exponent match: 6x^2 + 3x - 2x^2 + 8x = 4x^2 + 11x. Never merge an x^2 term with an x term.
To multiply two binomials, use FOIL (First, Outer, Inner, Last): (x + 4)(x + 3) = x^2 + 3x + 4x + 12 = x^2 + 7x + 12. Watch signs with subtraction: (x - 5)(x + 2) = x^2 + 2x - 5x - 10 = x^2 - 3x - 10. A frequent trap is squaring (x + 3) as x^2 + 9; it is actually x^2 + 6x + 9 because the middle term 2(x)(3) cannot be skipped. The same goes for a difference of squares: (x + 6)(x - 6) = x^2 - 36, where the Outer and Inner terms cancel exactly.
Quadratics: Set Equal to Zero First
A quadratic equation contains an x^2 term and is solved once it is in the form ax^2 + bx + c = 0. Many GED quadratics factor cleanly.
Solve x^2 + 7x + 12 = 0. Find two numbers that multiply to 12 and add to 7: that is 3 and 4. Factor to (x + 3)(x + 4) = 0, then set each factor to zero: x = -3 or x = -4.
If the trinomial does not factor with whole numbers, identify a, b, c and use the formula sheet's quadratic formula. Then interpret: a width or time that comes out negative is rejected. If a rectangle solves to x = -4 or x = 7, only 7 is a valid length. Choosing the right tool, factoring for clean trinomials and the formula otherwise, saves time within the 115-minute limit.
Test-Day Method Choice
Use this decision list to avoid wasted steps:
- For a system, use substitution when a variable is already alone, and elimination when adding or subtracting cancels a variable cleanly.
- For a quadratic, confirm it is set equal to zero before doing anything else; x^2 + 7x = -12 must become x^2 + 7x + 12 = 0 first.
- Try factoring first; if no whole-number pair multiplies to ac and adds to b, switch to the quadratic formula from the sheet.
- For polynomials, underline like terms before combining, and FOIL slowly when a minus sign is present.
- For every context answer, ask whether a negative or fractional solution is physically possible and discard the impossible one.
Solve the system: x + y = 14 and x - y = 2.
Factor x^2 + 9x + 20.
Expand (x - 5)(x + 2).