On the test
Advanced Math is about 35% of the Math section, around 6 to 7 questions in each 22-question module. Math modules mix all four domains and go from easier to harder, so these questions appear all through the module, with the hardest ones near the end.
13 to 15 of the 44 questions (about 6 or 7 in each module). All through the module, easier first and harder at the end.
How I approached it
A small story from Aime
For these questions I think about where two things meet. Imagine you walk from home to school on the road, and your friend walks from the market to school on a different path. You ask: where do we meet? There may be one place, two places, or no place at all if our paths never cross. A bus and a bicycle on the same road work the same way. Solving a system means finding the exact point where two paths share the same spot. When one path is a straight line and the other is a curve, there can be two meeting points. On the SAT, I first say: line and curve, so I make both equal and solve. Then I check every answer, because sometimes a meeting point is fake.
Even the hard questions in this skill are the same two steps repeated: make the two sides equal, then solve a quadratic. You can learn these steps, and Desmos will help you check.
What is inside this part
Equations with only x where x is squared, under a root, or in a fraction, and you find the x-values that work.
quadratic equations · square root equations and extraneous answers · equations with x in a fraction · sum or product of solutions
Two equations in x and y, one straight and one a U shape, solved by setting them equal.
finding the intersection points · testing ordered pairs · finding a missing constant so there is exactly one solution
A line and an equation like x² + y² = r², solved by substituting the line into the circle.
finding the crossing points · tangent lines · number of intersection points
Deciding if a system or equation has 0, 1 or 2 solutions, usually with the discriminant.
discriminant b² - 4ac · tangent line · finding the value of a constant
How to know a question is from this part
You get one equation with x² or a square root, or two equations (one has x² or x² + y²) and are asked for a solution, a point, the number of solutions, or a constant.
- The words 'system of equations' where one equation has x² or y².
- 'Exactly one solution', 'no real solutions', or 'tangent'.
- 'Which ordered pair (x, y) is a solution to the system?'
- A square root sign with x inside, or x in a bottom of a fraction.
- 'What is the sum of all solutions?' or 'What is the positive solution?'
The lesson
1. What these questions are about
This skill has two families. The first is a nonlinear equation in one variable: an equation with only x, but x appears as x², under a square root, or in a fraction. For example x² - 5x + 6 = 0 or √(x + 5) = 4.
The second is a system of equations in two variables: two equations with x and y, solved together. In this skill one equation is usually a line (like y = 2x + 5) and the other is a curve (like y = x² - 3, or a circle x² + y² = 25).
In both families you are looking for the values that make every equation true at the same time.
2. Solving a quadratic equation
First, move everything to one side so the other side is 0. Then you have three tools. Factoring: find two numbers that multiply to c and add to b. Quadratic formula: x = (-b ± √(b² - 4ac)) ÷ (2a). Square roots: if (x - 3)² = 16, then x - 3 = 4 or x - 3 = -4.
A quadratic usually has two solutions. Always say 'two' in your head, and then read the question: sometimes it wants the larger one, the positive one, or their sum.
A quick fact: for x² + bx + c = 0, the sum of the solutions is -b and the product is c. This is useful when the question asks for a sum or a product.
Solve 2x² - 8x - 10 = 0. What is the positive solution?
Show the steps
- Divide every term by 2: x² - 4x - 5 = 0.
- Factor: two numbers that multiply to -5 and add to -4 are -5 and 1. So (x - 5)(x + 1) = 0.
- x = 5 or x = -1. The positive solution is 5.
5
3. Square roots and extraneous solutions
When x is under a square root, you square both sides to remove the root. Squaring can create a fake answer called an extraneous solution. So after solving you must put each answer back into the original equation.
Also remember: a square root sign gives only the positive root. So √(something) = x needs x to be 0 or more.
Solve √(2x + 3) = x.
Show the steps
- Square both sides: 2x + 3 = x².
- Move: x² - 2x - 3 = 0. Factor: (x - 3)(x + 1) = 0. So x = 3 or x = -1.
- Check x = 3: √9 = 3. Works.
- Check x = -1: √1 = 1, but x = -1. Does not work. It is extraneous.
Only x = 3.
4. Systems: a line and a parabola
A solution of a system is a point (x, y) that is on both graphs. The method I trust is substitution: since both equations say what y equals, set the two right sides equal.
Take y = x² - 3 and y = 2x + 5. Set them equal: x² - 3 = 2x + 5. Move everything to one side: x² - 2x - 8 = 0. Factor: (x - 4)(x + 2) = 0. So x = 4 or x = -2.
Do not stop at x. Put each x into the simpler equation (the line) to get y. x = 4 gives y = 13. x = -2 gives y = 1. The solutions are (4, 13) and (-2, 1).
Which point is a solution of y = x² - 3 and y = 2x + 5, if the choices are (2, 9), (4, 13), (4, 5)?
Show the steps
- Test each point in both equations.
- (4, 13): 4² - 3 = 13 and 2(4) + 5 = 13. Both true.
- (2, 9): 2² - 3 = 1, not 9. (4, 5): 4² - 3 = 13, not 5.
(4, 13)
5. Systems: a line and a circle
The equation of a circle with center at the origin is x² + y² = r², where r is the radius. For example x² + y² = 25 is a circle of radius 5.
With a line y = x + 1, substitute the line into the circle: x² + (x + 1)² = 25. Expand: x² + x² + 2x + 1 = 25, so 2x² + 2x - 24 = 0, so x² + x - 12 = 0, so (x + 4)(x - 3) = 0. Then x = -4 or x = 3.
Put them in the line: x = 3 gives y = 4. x = -4 gives y = -3. The points are (3, 4) and (-4, -3). Check one: 9 + 16 = 25. Good.
Be careful with (x + 1)². It is x² + 2x + 1, not x² + 1. This is the most common mistake in this skill.
For the points above, what is the greatest value of x + y?
Show the steps
- (3, 4) gives 7.
- (-4, -3) gives -7.
7
6. How many solutions? Use the discriminant
A line and a curve can meet in 2 points, 1 point (the line just touches, which is called tangent), or 0 points. After you substitute, you get a quadratic. Its discriminant b² - 4ac tells the story: positive means 2 solutions, zero means exactly 1, negative means none.
So when a question says 'exactly one solution', 'tangent', or 'no real solutions', set the discriminant to 0 or less than 0 and solve for the missing number.
Example: y = x² - 2x + 3 and y = 2x + k. Set equal: x² - 4x + 3 - k = 0. Discriminant: 16 - 4(3 - k) = 4 + 4k. For exactly one solution: 4 + 4k = 0, so k = -1.
7. Rational and other one-variable equations
Sometimes x is in a fraction, like 12 ÷ (x - 3) = x. Multiply both sides by the bottom to clear the fraction, then solve the quadratic. Never allow an x that makes the bottom 0, because you cannot divide by zero.
Sometimes the equation is already a perfect square, like (x - 4)² = 25. Take the square root on both sides and remember the plus and minus: x - 4 = 5 or x - 4 = -5, so x = 9 or x = -1.
8. Common traps
Trap 1: finding x and forgetting y when the question asks for the point or for y.
Trap 2: expanding (x + 1)² wrongly.
Trap 3: not checking square-root equations for extraneous answers.
Trap 4: giving the number of solutions as 2 when the discriminant is 0 (it is 1).
Trap 5: using the wrong sign when moving terms. Move everything to one side and re-check the signs of a, b, c.
See it
A line can cross a circle at two points. Each crossing point is one solution (x, y) of the system.
A line can meet a parabola twice, touch it once (tangent), or miss it. The discriminant tells you which.
- 1Write both equations with y alone (or x alone)
- 2Set the two expressions equal (substitution)
- 3Move everything to one side to make a quadratic = 0
- 4Factor or use the formula to get x
- 5Put each x into the line to get y; check
| b² - 4ac | Number of real solutions | Line and curve picture |
|---|---|---|
| positive | 2 | line cuts the curve twice |
| zero | 1 | line just touches (tangent) |
| negative | 0 | line never meets the curve |
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