SAT Math: Nonlinear equations and systems

Nonlinear equations in one variable and systems in two variables.

Advanced Math · Nonlinear equations in one variable and systems of equations in two variables
Both modules
20 practice questions
Calculator allowed (Desmos)

Practise nonlinear equations and systems →Adaptive SAT module

What the test covers

Quadratics (factor, formula, discriminant), radical and rational equations with extraneous roots
Line-parabola systems

Key ideas

Quadratics: factorise, complete the square or use \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\). The sum of the roots is \(-\tfrac ba\).
The discriminant \(b^2-4ac\) decides whether there are 2, 1 or 0 real solutions.
Radical and rational equations can produce extraneous solutions: always check in the original equation.
A line and a parabola meet at 0, 1 or 2 points: substitute and use the discriminant.

Common mistakes

Squaring \(\sqrt{x+7}=x-5\) creates a false root; check both.
A denominator can never be zero.

Do it in Desmos

Graph both sides and count or read the intersections.

IB link: IB SL 2.7.

Worked example

What is the sum of the solutions to \(2x^2-7x+3=0\)?

\(\frac{3}{2}\)
\(\frac{7}{2}\)
\(7\)
\(-\frac{7}{2}\)

Answer: B: \(\frac{7}{2}\)

\((2x-1)(x-3)=0\), so \(x=\tfrac12\) or \(3\); the sum is \(\tfrac72\).

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.

Question 1

In the equation \(x^2-10x+c=0\), \(c\) is a constant. The equation has exactly one real solution. What is the value of \(c\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 25

One real solution means the discriminant is zero: \(100-4c=0\), \(c=25\).

Question 2

What are all the solutions to \(\sqrt{x+7}=x-5\)?

There are no solutions
\(2\) and \(9\)
\(9\) only
\(2\) only

Show the answer and solution

Answer: C: \(9\) only

Squaring: \(x+7=x^2-10x+25\), \(x^2-11x+18=0\), \((x-2)(x-9)=0\). Check: \(x=9\): \(\sqrt{16}=4=9-5\) ✓. \(x=2\): \(\sqrt9=3\ne-3\) ✗ (created by squaring). Only 9.

Question 3

How many solutions does the system \(y=x^2-4x+6\) and \(y=2x-3\) have?

Exactly two
Zero
Infinitely many
Exactly one

Show the answer and solution

Answer: D: Exactly one

\(x^2-4x+6=2x-3\) gives \(x^2-6x+9=(x-3)^2=0\): one solution, \((3,3)\). The line is tangent to the parabola.

Question 4

What value of \(x\) satisfies \(\dfrac{3}{x-1}=\dfrac{2}{x+4}\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: -14

Cross-multiply: \(3(x+4)=2(x-1)\), \(3x+12=2x-2\), \(x=-14\) (which does not make a denominator zero).

Question 5

The line \(y=kx-1\), where \(k\) is a positive constant, intersects the parabola \(y=x^2+3\) at exactly one point. What is the value of \(k\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 4

\(x^2+3=kx-1\) gives \(x^2-kx+4=0\). Exactly one intersection: \(k^2-16=0\), \(k=\pm4\). As \(k>0\), \(k=4\).

Question 6

What is the sum of the solutions to \(4 x^{2} - 8 x - 12=0\)?

\(-3\)
\(2\)
\(12\)
\(-2\)

Show the answer and solution

Answer: B: \(2\)

The solutions are \(3\) and \(-1\) (factor, or use sum \(=-\tfrac{b}{a}\)). Their sum is \(2\).

Question 7

What is the sum of the solutions to \(3 x^{2} - 7 x + 2=0\)?

\(-2\)
\(- \frac{7}{3}\)
\(\frac{7}{3}\)
\(\frac{2}{3}\)

Show the answer and solution

Answer: C: \(\frac{7}{3}\)

The solutions are \(2\) and \(\frac{1}{3}\) (factor, or use sum \(=-\tfrac{b}{a}\)). Their sum is \(\frac{7}{3}\).

Question 8

What is the sum of the solutions to \(3 x^{2} + 3 x - 6=0\)?

\(6\)
\(1\)
\(-2\)
\(-1\)

Show the answer and solution

Answer: D: \(-1\)

The solutions are \(1\) and \(-2\) (factor, or use sum \(=-\tfrac{b}{a}\)). Their sum is \(-1\).

Question 9

What is the sum of the solutions to \(4 x^{2} - 14 x + 6=0\)?

\(\frac{7}{2}\)
\(-6\)
\(- \frac{7}{2}\)
\(\frac{3}{2}\)

Show the answer and solution

Answer: A: \(\frac{7}{2}\)

The solutions are \(3\) and \(\frac{1}{2}\) (factor, or use sum \(=-\tfrac{b}{a}\)). Their sum is \(\frac{7}{2}\).

Question 10

What is the sum of the solutions to \(2 x^{2} + 3 x - 2=0\)?

\(-1\)
\(2\)
\(- \frac{3}{2}\)
\(\frac{3}{2}\)

Show the answer and solution

Answer: C: \(- \frac{3}{2}\)

The solutions are \(-2\) and \(\frac{1}{2}\) (factor, or use sum \(=-\tfrac{b}{a}\)). Their sum is \(- \frac{3}{2}\).

Question 11

In the equation \(x^2-20x+c=0\), \(c\) is a constant. The equation has exactly one real solution. What is the value of \(c\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 100

One real solution means the discriminant is zero: \(20^2-4c=0\), so \(c=100\).

Question 12

In the equation \(x^2-6x+c=0\), \(c\) is a constant. The equation has exactly one real solution. What is the value of \(c\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 9

One real solution means the discriminant is zero: \(6^2-4c=0\), so \(c=9\).

Question 13

In the equation \(x^2-18x+c=0\), \(c\) is a constant. The equation has exactly one real solution. What is the value of \(c\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 81

One real solution means the discriminant is zero: \(18^2-4c=0\), so \(c=81\).

Question 14

In the equation \(x^2-22x+c=0\), \(c\) is a constant. The equation has exactly one real solution. What is the value of \(c\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 121

One real solution means the discriminant is zero: \(22^2-4c=0\), so \(c=121\).

Question 15

In the equation \(x^2-16x+c=0\), \(c\) is a constant. The equation has exactly one real solution. What is the value of \(c\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 64

One real solution means the discriminant is zero: \(16^2-4c=0\), so \(c=64\).

Question 16

The line \(y=kx-5\), where \(k\) is a positive constant, intersects the parabola \(y=x^2+4\) at exactly one point. What is the value of \(k\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 6

\(x^2+4=kx-5\) gives \(x^2-kx+9=0\). Exactly one intersection: \(k^2-36=0\), \(k=\pm6\). As \(k>0\), \(k=6\).

Question 17

The line \(y=kx-1\), where \(k\) is a positive constant, intersects the parabola \(y=x^2+24\) at exactly one point. What is the value of \(k\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 10

\(x^2+24=kx-1\) gives \(x^2-kx+25=0\). Exactly one intersection: \(k^2-100=0\), \(k=\pm10\). As \(k>0\), \(k=10\).

Question 18

The line \(y=kx-1\), where \(k\) is a positive constant, intersects the parabola \(y=x^2+8\) at exactly one point. What is the value of \(k\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 6

\(x^2+8=kx-1\) gives \(x^2-kx+9=0\). Exactly one intersection: \(k^2-36=0\), \(k=\pm6\). As \(k>0\), \(k=6\).

Question 19

The line \(y=kx-1\), where \(k\) is a positive constant, intersects the parabola \(y=x^2+15\) at exactly one point. What is the value of \(k\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 8

\(x^2+15=kx-1\) gives \(x^2-kx+16=0\). Exactly one intersection: \(k^2-64=0\), \(k=\pm8\). As \(k>0\), \(k=8\).

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