Zeros, roots, x-intercepts, and solutions all name the same thing: the inputs where a function equals zero. The SAT swaps the vocabulary on purpose. When a quadratic is given in factored form, those values are immediately readable, and they also unlock the axis of symmetry and the vertex.
Set each factor equal to zero and solve, and remember the axis of symmetry is halfway between the two zeros.
If f of x equals the quantity x minus 3 times the quantity x plus 5, then the function is zero when either factor is zero, so the zeros are x equals 3 and x equals negative 5. On a graph, those are the points where the parabola crosses the horizontal axis, written as the points with coordinates 3 and 0 and coordinates negative 5 and 0. A question asking for an x-intercept may want the ordered pair or just the number, so read the format of the choices.
The sign flip is constant and worth saying out loud each time: the factor x minus 3 produces the zero positive 3, and the factor x plus 5 produces the zero negative 5. A factor of 2x minus 7 produces a zero of seven halves, because you set 2x minus 7 equal to zero and solve rather than reading the 7 directly.
A parabola is symmetric about a vertical line through its vertex, and that line sits exactly halfway between the zeros. With zeros at 3 and negative 5, the midpoint is negative 1, so the axis of symmetry is x equals negative 1. Substituting negative 1 into the function gives the y-coordinate of the vertex, which is negative 4 times 4, or negative 16 in this example.
Which form to want depends on the question. Factored form shows the zeros, vertex form shows the maximum or minimum, and standard form shows the y-intercept, which is the constant term. Many SAT questions ask which equivalent equation displays a stated feature as a constant or coefficient, and the answer is whichever form puts that feature in plain sight.
In context, zeros carry meaning. If h of t models the height of a thrown object and factors as negative 5 times t times the quantity t minus 6, the zeros at t equals 0 and t equals 6 are the launch time and the landing time. The vertex then sits at t equals 3, halfway between, which is when the object reaches its greatest height.
Reporting the numbers inside the parentheses as the zeros, so the quantity x plus 5 gets called a zero of 5 instead of negative 5.
A population doubles every 3 years. If it is 1000 now, what will it be in 12 years?
A patient receives 500 milligrams of a drug, and each hour 8% of the drug still present is cleared. To the nearest milligram, how much remains after 6 hours?
Why: Each period multiplies by 1 - 8/100 = 0.92, so the amount is 500(0.92)^6 ≈ 303. The value 460 applies the decrease a single time, which leaves far too much. Answering 330 means you counted one period too few.
They describe the same values from different angles. Zeros and roots are the inputs that make the function equal zero. X-intercepts are the points where the graph crosses the horizontal axis. The numbers match, but an intercept is a point while a zero is a single value.
Then the two factors are identical, as in the quantity x minus 4 squared, and the parabola touches the x-axis at exactly one point rather than crossing it. That point is the vertex, and the discriminant of the equation is zero.