Using the Discriminant for Equal Roots

The discriminant $b^2 - 4ac$ tells you the nature of a quadratic’s roots without solving it. The skill examiners test is translating the English into the right condition, then solving the resulting equation in the unknown constant. “Equal roots” always means $b^2 - 4ac = 0$.

The equation $x^2 + (k + 3)x + (2k + 3) = 0$ has two equal roots. Find the possible values of the constant $k$. [5]

The working

Step 1, identify $a$, $b$, $c$ (careful: $b$ and $c$ contain $k$): $a = 1, \qquad b = k + 3, \qquad c = 2k + 3$

Step 2, apply the equal-roots condition $b^2 - 4ac = 0$: $(k + 3)^2 - 4(1)(2k + 3) = 0 \quad \text{(M1 for the condition, M1 for substituting)}$

Step 3, expand and simplify into a quadratic in $k$: $k^2 + 6k + 9 - 8k - 12 = 0 \;\Rightarrow\; k^2 - 2k - 3 = 0 \quad \text{(A1)}$

Step 4, solve by factorising: $(k - 3)(k + 1) = 0 \;\Rightarrow\; k = 3 \quad \text{or} \quad k = -1 \quad \text{(M1, A1)}$

Where the marks are won and lost

• Getting $b = k + 3$ and $c = 2k + 3$ as whole expressions into the discriminant is the first method mark. Dropping a bracket, for example writing $k + 3^2$ instead of $(k+3)^2$, corrupts everything after.
• $(k + 3)^2$ expands to $k^2 + 6k + 9$ (with the middle term). Forgetting $6k$ is the usual error.
• Both values of $k$ are required. A discriminant condition often produces a quadratic in the constant, so expect two answers and check neither is to be rejected.

Common mistakes

• Using $b^2 - 4ac > 0$ (distinct roots) when the question says equal.
• Expanding $(k + 3)^2$ as $k^2 + 9$.
• Solving the original equation for $x$ instead of forming the condition on $k$.

Full method: Discriminant & Nature of Roots notes. Topic home: Quadratic Functions pillar.