Completing the square rewrites a quadratic so its turning point is visible. The discriminant tells you how many solutions to expect before you solve anything.
Write x² + bx + c as (x + b/2)² + (c − b²/4). The graph then has its vertex at (−b/2, c − b²/4). For a positive x² coefficient this is the minimum point.
For ax² + bx + c = 0, the discriminant is Δ = b² − 4ac. It sits under the root in the quadratic formula x = (−b ± √Δ) ÷ 2a.
Write x² + 6x + 5 in completed-square form.
If the discriminant is negative, the quadratic has:
The graph of y = (x − 2)² + 3 has its minimum at:
The quadratic x² − 4x + 4.
What is the discriminant?
Enter an exact answer, without rounding.The quadratic 2x² + 3x − 2.
What is the discriminant?
Enter an exact answer, without rounding.The quadratic x² + 6x + 5 = (x + 3)² − 4.
What is its minimum value?
Enter an exact answer, without rounding.Solve (x + 3)² = 4 by taking square roots. One root is x = −1.
What is the other root?
Enter an exact answer, without rounding.The equation x² + kx + 9 = 0 has a repeated root.
What positive value of k is needed?
Enter an exact answer, without rounding.The discriminant of ax² + bx + c is:
Δ = 0 means the graph:
x² − 2x − 3 in completed-square form is:
The quadratic formula gives x =
Which quadratic has no real roots?
Functions can be chained together, and many can be run backwards. Both ideas are central to A-Level calculus and modelling.
fg(x) means do g first, then f: fg(x) = f(g(x)). The order matters: fg and gf are usually different.
The inverse f⁻¹ undoes f. To find it, write y = f(x), swap x and y, then solve for y. The graph of f⁻¹ is the reflection of f in the line y = x. The domain of f⁻¹ is the range of f.
In fg(x), which function is applied first?
The graph of f⁻¹ is a reflection of the graph of f in the line:
What is the inverse of f(x) = x + 5?
f(x) = 2x + 3 and g(x) = x².
What is fg(2)?
Enter an exact answer, without rounding.f(x) = 2x + 3 and g(x) = x².
What is gf(2)?
Enter an exact answer, without rounding.f(x) = 2x + 3, so f⁻¹(x) = (x − 3) ÷ 2.
What is f⁻¹(11)?
Enter an exact answer, without rounding.f(x) = 3x − 1.
What is f⁻¹(8)?
Enter an exact answer, without rounding.f(x) = x + 4 and g(x) = 3x.
What is fg(1)?
Enter an exact answer, without rounding.If f(x) = x² on the domain x ≥ 0, then f⁻¹(x) is:
The domain of f⁻¹ equals the:
For f(x) = 2x, f⁻¹(x) is:
Is it always true that fg(x) = gf(x)?
Which functions have inverses?
Questions from every topic in this course. Pick an answer for each one, then hit Check My Answers.
The line y = 2x + c is tangent to y = x². Substituting gives x² − 2x − c = 0. Then c equals:
x² + 6x + 2 in completed-square form is:
The discriminant of x² − 4x + 4 is:
If b² − 4ac < 0, the quadratic equation has:
f(x) = x + 1 and g(x) = x². Then gf(x) is:
f(x) = 2x + 1 and g(x) = x². fg(3) is:
The inverse of f(x) = 3x − 6 is:
f⁻¹(f(x)) equals: