Questions: Quadratic Formula Review and Applications

Question 1

For x² + 4x + 13 = 0, you calculate the discriminant as 16 − 52 = −36. Before finishing the calculation, what can you immediately conclude?

• A. There is one repeated real solution because the discriminant is a perfect square
• B. There are no solutions at all — the equation is unsolvable
• C. There are two complex conjugate solutions — the parabola does not cross the x-axis
• D. There are two real solutions because 36 is positive

Answer: (C)

A negative discriminant (D < 0) means the square root in the formula is the square root of a negative number, which is imaginary. The result is two complex conjugate solutions of the form p ± qi. Graphically, the parabola lies entirely above (or below) the x-axis and never crosses it. The discriminant delivers this verdict before you finish computing — that is its power. Option D confuses the absolute value of D with its sign.

Question 2

For 3x² − 6x + 3 = 0, the discriminant equals b² − 4ac = 36 − 36 = 0. What does this tell you geometrically?

• A. The parabola crosses the x-axis at two distinct points
• B. The vertex of the parabola touches the x-axis exactly once — one repeated real root
• C. The parabola is entirely below the x-axis with no real roots
• D. The discriminant being zero means the equation has no solution

Answer: (B)

When D = 0, the ± in the formula adds and subtracts zero, collapsing to a single value: x = −b/(2a) = 6/6 = 1. Geometrically, the vertex of the parabola sits exactly on the x-axis — it touches but does not cross. This is called a repeated (or double) root. Students sometimes confuse D = 0 with no solution, but one solution is not the same as no solution.

Question 3

The quadratic formula is expected to be memorized as an independent rule because it can seldom be derived from techniques already learned in algebra.

• T. True
• F. False

Answer: False

The quadratic formula is the direct result of completing the square on the general form ax² + bx + c = 0. Every step is the same completing-the-square procedure applied symbolically. Knowing the derivation means you understand where the formula comes from, can re-derive it if needed, and are less likely to misremember it. It is not a separate fact — it is a packaged version of something you already know how to do.

Question 4

When the discriminant equals zero, the quadratic has exactly one real solution, corresponding to the vertex of the parabola touching the x-axis.

• T. True
• F. False

Answer: True

D = 0 means √D = 0, so the ± term vanishes and both 'solutions' from the ± collapse to the single value x = −b/(2a). This is the x-coordinate of the vertex, confirming the geometric picture: the parabola is tangent to the x-axis at exactly one point. This solution is called a repeated root because it counts algebraically as a root of multiplicity 2.

Question 5

Why does −b in the quadratic formula require special care when b is already negative, and what type of error commonly results from mishandling this sign?

Model answer: The formula uses −b, which means you negate whatever b is. If b = −5, then −b = −(−5) = +5. If you forget the negation and write b instead of −b, you get the wrong value for the numerator. The most common error is substituting b = −5 and writing −5 in the formula instead of +5, effectively computing the formula with the wrong sign on the linear term. Careful substitution — writing out −(−5) explicitly before simplifying — prevents this mistake.

Sign errors in the quadratic formula account for a large fraction of wrong answers on assessments. The issue is compounded because the error affects both terms in the ± simultaneously, and the resulting 'solutions' may still look plausible. Checking solutions by substituting back into the original equation is the reliable safeguard.