The key idea
For ax2+bx+c=0, the discriminant b2−4ac tells you the number of real solutions: positive means two, zero means one, negative means none. A line is tangent to a parabola when substituting gives a quadratic with discriminant 0.
The trap
Forgetting to move every term to one side before reading a, b, and c.
Can Desmos do it?
For a fixed equation, graphing shows the number of x-intercepts. When a constant is unknown, set the discriminant equal to 0 instead of hunting with a slider.
College Board skill: Nonlinear equations in one variable and systems of equations in two variables · Advanced Math
One question from each level. The drill has 12 at every level, and moves you up after four right in a row.
Level 1 · Warm-up
How many real solutions does the equation 4x2−3x+4=0 have?
- ANo real solutions
- BExactly one real solution
- CExactly two real solutions
- DInfinitely many solutions
Show the answer
Answer: AThe discriminant is b2−4ac=(−3)2−4(4)(4)=−55, which is negative, so the equation has no real solutions.
Level 2 · Test level
In the equation x2+kx+4=0, k is a positive constant. If the equation has exactly one real solution, what is the value of k?
Show the answer
Answer: 4Exactly one solution means the discriminant is 0: k2−4(1)(4)=0, so k2=16 and, since k>0, k=4.
Level 3 · Beyond the test
In the xy-plane, the line y=−3x+b, where b is a constant, intersects the parabola y=x2−7x at exactly one point. What is the value of b?
Show the answer
Answer: -4Set the expressions equal: x2−7x=−3x+b, or x2−4x−b=0. One intersection means the discriminant is 0: (−4)2−4(0−b)=0, so 0−b=4 and b=−4.
Practice all 36 questions
Answer in the drill to get instant feedback, climb the levels, and feed your misses into Tailored practice. Free, no account.
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