2.3 Quadratic functions, the discriminant and completing the square — Practise

Work through these on your own. For each item, decide which feature the question wants — a root, a vertex, or the number of roots — and choose the form of the quadratic that reads that feature off.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

Write $x^2 + 6x + 2$ in the form $(x + a)^2 + b$ .
Write $2x^2 - 8x + 5$ in the form $a(x + p)^2 + q$ .
Complete the square for $x^2 - 10x + 3$ , and state the minimum value of the expression.
Work out the discriminant of $3x^2 + 2x + 5$ and use it to state how many real roots the equation $3x^2 + 2x + 5 = 0$ has.
Show that $x^2 - 4x + 4 = 0$ has a repeated root, and find it.
Work out the discriminant of $x^2 - 5x + 6$ and state how many real roots the equation has.
Find the coordinates of the vertex of $y = x^2 - 6x + 5$ .
Show that $x^2 + 4x + 7$ is positive for every value of $x$ , and state its minimum value.
Factorise $x^2 - 7x + 12$ .
The equation $x^2 + kx + 9 = 0$ has a repeated root. Find the possible values of $k$ .
Show that the equation $2x^2 - 4x + 5 = 0$ has no real roots.
For $y = x^2 - 2x - 3$ , say which feature you would find each way. (a) the roots (b) the vertex.

Answers and concise solutions

1. Completed square

Answer: Half of $6$ is $3$ , and $\left(x + 3\right)^2 = x^2 + 6x + 9$ . So $x^2 + 6x + 2 = \left(x + 3\right)^2 - 7$ : the $+9$ is removed by subtracting $7$ .

2. Completed square

Answer: Take out the factor $2$ from the first two terms: $2\left(x^2 - 4x\right) + 5$ . Complete the square inside the bracket: $x^2 - 4x = \left(x - 2\right)^2 - 4$ . So $2\left(x^2 - 4x\right) + 5 = 2\left(x - 2\right)^2 - 8 + 5 = 2\left(x - 2\right)^2 - 3$ .

3. The completed square and the minimum

Answer: $x^2 - 10x + 3 = \left(x - 5\right)^2 - 22$ . A square is never negative, so the smallest value occurs when $\left(x - 5\right)^2 = 0$ , at $x = 5$ . The minimum value is $-22$ .

4. The discriminant and the number of roots

Answer: The discriminant is $b^2 - 4ac = 2^2 - 4 \times 3 \times 5 = 4 - 60 = -56$ . It is negative, so the equation has no real roots.

5. The discriminant and the root

Answer: The discriminant is $(-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0$ , so there is exactly one repeated real root. Factorising gives $\left(x - 2\right)^2 = 0$ , so the repeated root is $x = 2$ .

6. The discriminant and the number of roots

Answer: The discriminant is $(-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$ . It is positive, so there are two distinct real roots. They are $x = 2$ and $x = 3$ , since $x^2 - 5x + 6 = \left(x - 2\right)\left(x - 3\right)$ .

7. The vertex

Answer: Complete the square: $x^2 - 6x + 5 = \left(x - 3\right)^2 - 4$ . The square is smallest when $x = 3$ , where it is $0$ . Substituting $x = 3$ gives $y = -4$ , so the vertex is at $(3, -4)$ and the curve opens upwards.

8. The completed square and the minimum

Answer: $x^2 + 4x + 7 = \left(x + 2\right)^2 + 3$ . A square is never negative, so the expression is at least $3$ . Since $3 > 0$ , the expression is positive for every $x$ , and its minimum value is $3$ at $x = -2$ .

9. Factorised

Answer: Look for two numbers that multiply to $12$ and add to $-7$ . The numbers are $-3$ and $-4$ , so $x^2 - 7x + 12 = \left(x - 3\right)\left(x - 4\right)$ .

10. The values of $k$

Answer: A repeated root needs a discriminant of zero: $k^2 - 4 \times 1 \times 9 = 0$ , so $k^2 = 36$ and $k = 6$ or $k = -6$ . Both values give $\left(x \pm 3\right)^2 = 0$ .

11. The discriminant and the conclusion

Answer: The discriminant is $(-4)^2 - 4 \times 2 \times 5 = 16 - 40 = -24$ . It is negative, and a negative discriminant means no real roots. Completing the square also shows this: $2x^2 - 4x + 5 = 2\left(x - 1\right)^2 + 3$ , which is always at least $3$ .

12. Which method, and why

Answer: (a) For the roots, factorise or use the formula: $x^2 - 2x - 3 = \left(x - 3\right)\left(x + 1\right)$ , so the roots are $x = 3$ and $x = -1$ . (b) For the vertex, complete the square: $x^2 - 2x - 3 = \left(x - 1\right)^2 - 4$ , so the vertex is at $(1, -4)$ . Factorising reads the roots straight off; completing the square reads the vertex straight off.