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Factor, compose functions, read slopes and locate extrema without calculus.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
Use equivalent algebraic and coordinate representations to compute roots, function values, slopes and constrained extrema.
Review distributive multiplication, exponent rules and linear equations. An expression such as x squared - 5x + 6 has a value for every real input; an equation setting it equal to zero asks which inputs make that value zero. A graph shows input-output pairs, with x normally horizontal and y vertical. Coordinates are ordered: (2, 5) differs from (5, 2). In a relation that is a function, each allowed input has exactly one output, although different inputs may share the same output.
| Term | What it means |
|---|---|
| Root or zero | An input at which a function has output zero. |
| Slope | Change in vertical coordinate divided by change in horizontal coordinate for a nonvertical line. |
| Composition | Using one function's output as another function's input. |
| Vertex | The turning point of a parabola, where its maximum or minimum occurs. |
An expression can have several equivalent forms, each making a different feature visible. A factored quadratic exposes zeros. A completed square exposes a maximum or minimum. Slope-intercept form exposes a line's rate of change and vertical intercept. A function rule exposes how an input becomes an output. The question determines which form is useful; expanding everything immediately can hide the very structure you need.
Equivalence also carries a domain. The rational expression (x squared - 4)/(x - 2) simplifies to x + 2 only where the original denominator is nonzero. Its value at x = 2 remains undefined. A shortened expression is a tool for evaluating permitted inputs, not permission to add new ones. Keep restrictions beside your working so they survive every transformation.
Another way: steps
Identify the target and domain; choose a useful form; carry out the operation; verify in the original representation.
When a product equals zero, at least one factor equals zero. Thus x squared - x - 12 = 0 becomes (x - 4)(x + 3) = 0, producing roots four and negative three. The zero-product rule applies to a product equal to zero, not to an arbitrary sum or a product equal to five. Move all terms to one side before using it. Check the factorization by multiplying it out. If simple integer factors are unavailable, completing the square or the quadratic formula gives another route; a quadratic need not have integer roots.
For ax squared + bx + c = 0 with a nonzero, the quadratic formula has numerator -b plus or minus the square root of b squared - 4ac, divided by 2a. The discriminant b squared - 4ac is positive for two distinct real roots, zero for one repeated real root, and negative for no real roots. GMAT quantitative problems use real numbers unless the question defines otherwise; do not invent a real square root of a negative discriminant. Parenthesize the entire numerator when entering the formula into a calculator.
For (x - 1)(x - 5) < 0, the product is negative when the factors have opposite signs, which occurs between one and five. At either root the product is zero, so a strict inequality excludes them. Outside the roots both factors have the same sign and the product is positive. Test one point in each interval if needed, then reason that a continuous polynomial cannot change sign without crossing a root. If a factor is squared, its sign does not reverse at that root; inspect the factors rather than memorizing that every quadratic is negative between any two listed values.
Rewrite x squared + 8x + 21 as (x + 4) squared + 5. A real square is nonnegative, so the expression is at least five, attained when x = -4. For a negative leading coefficient, the corresponding form gives a maximum instead. If the domain restricts x to a positive interval or to integers, check whether the vertex is permitted. A mathematical minimum over all real numbers may not be feasible in the stated problem. No differentiation is needed; the nonnegativity of a square supplies the bound directly.
If f(t) = t squared - 2t, f(-3) means substitute -3 everywhere t appears: (-3) squared - 2(-3). It does not mean multiply f by negative three. For f(g(2)), first compute g(2), then feed that value into f. Reversing the order generally changes the result. A piecewise rule requires selecting the branch based on the input before evaluating. If the boundary belongs to one branch through a less-than-or-equal sign, do not choose the other branch merely because its formula seems easier.
For points (x1, y1) and (x2, y2), use slope (y2 - y1)/(x2 - x1), taking both differences in the same order. A horizontal line has slope zero because its vertical change is zero. A vertical line has undefined slope because horizontal change is zero; it cannot be represented as y = mx + b with a finite m. Distinct parallel nonvertical lines have equal slopes, while perpendicular nonvertical lines have slopes whose product is negative one. Handle a horizontal-vertical pair separately instead of trying to take the reciprocal of zero.
For y = mx + b, b is the y-intercept because x = 0 there. To find the x-intercept, set y = 0 and solve, provided the equation permits an isolated intercept. The distance between two points follows from a right triangle with horizontal and vertical legs: the square root of the sum of the squared coordinate differences. The midpoint averages each coordinate separately. A line inequality such as y > 2x + 1 selects points above the line and excludes the boundary; testing one point can identify the correct side of a coordinate region.
Substitute the chosen value into the original function or equation, then verify its domain and the quantity actually requested by the question.
A business model might express profit as a negative quadratic in the number of units sold. Completing the square reveals the greatest modeled profit and the quantity at which it occurs. Before reporting the vertex, check whether that quantity is a nonnegative integer and whether the model's stated range permits it. A fractional unit or an output above production capacity may require evaluating the nearest feasible alternatives instead.
The algebra establishes a result within the model. It does not prove that actual market demand or costs follow the formula indefinitely. Report the modeled optimum with its assumptions intact. This is the same discipline used in GMAT quantitative comparisons: a valid calculation cannot justify a conclusion outside the stated domain.
For review, solve the same kind of task from a different representation, such as a factored expression or a small table. Compare what each representation reveals quickly. The exercise is to choose an efficient method and verify feasibility, rather than expand every expression by habit or memorize a vertex value from an earlier question.
Cancelling a denominator does not make a forbidden input valid. A quadratic equation can have two, one or no real roots. Roots alone do not specify a quadratic inequality's solution set. A vertical line has undefined slope, not zero slope. Composing functions in reverse order can produce a different answer. A real-valued vertex may be excluded by an integer or contextual restriction.
Move all terms to one side.
x squared = x + 20 becomes x squared - x - 20 = 0.
The zero-product rule requires zero on one side.
Find the factor pair.
(x - 5)(x + 4) = 0.
The constants multiply to -20 and add to -1.
Set each factor to zero.
x = 5 or x = -4.
Either factor can make the product zero.
Check both original substitutions.
25 = 5 + 20; 16 = -4 + 20.
Both roots satisfy the original equation.
Identify coordinate differences.
The line passes through (1, -2) and (4, 7).
Keep the order of subtraction consistent.
Compute the signed slope.
(7 - (-2))/(4 - 1) = 3.
The line rises nine units over three horizontal units.
Substitute one point into line form.
-2 = 3(1) + b.
The slope is known and the intercept remains unknown.
Solve for the intercept.
b = -5, so y = 3x - 5.
Subtract three from both sides.
Check the second point.
At x = 4, the equation gives y = 7.
The line satisfies both supplied coordinates.
Read the expression and domain.
Minimize x squared - 5x + 9 for integer x.
An integer restriction may exclude the real vertex.
Complete the square.
(x - 2.5) squared + 2.75.
Expanding recovers the original expression.
Locate the real vertex.
The real minimum occurs at x = 2.5.
The square would be zero there.
Apply the integer condition.
The closest permitted inputs are 2 and 3.
Both are one half unit from the vertex.
Evaluate the nearest permitted inputs.
At either input the value is 3.
The square contributes 0.25 above the real lower bound.
Justify the global integer bound.
Every other integer is farther from 2.5 and has a larger squared distance.
Thus three is the constrained minimum, not 2.75.
Identify the inner function.
f(x)=2x+3; g(x)=x squared; compute g(f(-2)).
The inner input is negative two.
Compute the intermediate output.
f(-2) = -1.
Use this entire value as g's input.
Evaluate and distinguish the reverse order.
If x squared - 7x + 12 = 0, what is the greater solution?
If f(t) = 2t squared - 1, what is f(-3)?
What is the sum of the real solutions of x squared + 2x - 15 = 0?
For how many integers x is x squared - 5x + 6 < 0?
If f(x)=3x-4 and g(x)=x squared, what is f(g(2))?
What is the slope of the line through (2, 7) and (6, -1)?
A line has slope 3 and passes through (2, 11). What is its y-intercept?
A line is perpendicular to a line with slope -2/5. What is its slope? Enter the exact decimal.
What is the minimum value of x squared - 6x + 13 for real x?
If h(x)=(x squared - 9)/(x - 3), what is h(5)?
A product's modeled daily profit in dollars is P(q) = -q squared + 40q - 300, where q is the number sold. What is the greatest modeled profit for a nonnegative integer q?
Answer three reserved questions without hints. These cases are separate from lesson practice.
What is the greater real solution of x squared - 2x - 24 = 0?
If f(x)=x+2 and g(x)=4x, what is g(f(-1))?
A line passes through (-1, 4) and (3, 12). What is its y-intercept?
Tomorrow, compare f(g(x)) with g(f(x)) on one input, then solve a quadratic inequality using sign intervals. Explain why cancelling a denominator factor does not restore the excluded input.
18. Finish a composition in the correct order, step 3
g(-1)=1; f(g(-2)) would instead equal 11.
Function composition is generally not commutative.