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

Prerequisites: expanding brackets; factorising; the index laws from 2.1; substituting a value into an expression.

By the end of this section you can: write a quadratic in completed-square form; read off its vertex and its line of symmetry; use the discriminant to say how many real roots the equation has; and connect the completed-square sign to the shape of the curve.

1. Three forms of the same function

A quadratic function has the form $y = ax^2 + bx + c$, with $a \neq 0$. Its graph is a parabola. The same function can be written in three ways, and each way shows a different feature.

The three forms are equal for every value of $x$. Choosing a form is choosing which feature you want to read.

2. Completing the square

To complete the square for a monic quadratic (one with $a = 1$), take half of the coefficient of $x$ and write the square of $x$ plus that number.

Example 1. Write $x^2 + 8x + 5$ in completed-square form.

Half of $8$ is $4$, and

The expanded square has $16$ where the expression has $5$. Subtract the difference, $16 - 5 = 11$:

Watch the sign of the bracket: the square is $(x + 4)^2$ because the coefficient $+8$ is positive.

Example 2. Write $2x^2 - 12x + 7$ in the form $a(x + p)^2 + q$.

Take the factor $2$ out of the first two terms:

Complete the square inside the bracket. Half of $-6$ is $-3$, and $(x - 3)^2 = x^2 - 6x + 9$:

Substitute back and multiply the $-9$ by the outside factor $2$:

3. The vertex and the minimum

A square is never negative, so $(x + p)^2 \geq 0$ for every $x$. This is the key to the completed-square form.

For $y = a(x + p)^2 + q$ with $a > 0$:

• the smallest value of the bracket is $0$, at $x = -p$;
• the smallest value of $y$ is therefore $q$;
• the vertex (turning point) is $(-p, q)$;
• the line of symmetry is $x = -p$.

If $a < 0$ the parabola opens downwards, and the vertex gives the maximum instead. The $y$-intercept is always the value of $y$ when $x = 0$.

4. The discriminant

The roots of $ax^2 + bx + c = 0$ come from the quadratic formula

Everything about the roots is decided by the expression under the square root. It is called the discriminant, written $\Delta$:

The third row is the reason a negative discriminant is useful: it proves that a quadratic equation cannot be solved with real numbers, and it shows why.

5. Try it: the parabola explorer

Predict first. For $y = x^2 + bx + c$, the vertex moves as $b$ changes and the number of roots changes as $c$ changes. If you keep $c = 0$ and change $b$, will the vertex stay on the $y$-axis, or move? Write your prediction down before you touch the sliders.

Test. Move the sliders for $b$ and $c$. The panel reports the completed-square form, the vertex, and the discriminant.

Explain. For which values of $c$ does the parabola cross the $x$-axis twice? Find the value of $c$ at which the two crossings become one. Write two sentences linking the discriminant to what you see.

Watch out. Completing the square is an identity, and the sign of the bracket follows the sign of $b$.

For $x^2 - 6x$, half of $-6$ is $-3$, so the bracket is $(x - 3)^2$, not $(x + 3)^2$. Expand to check: $(x - 3)^2 = x^2 - 6x + 9$, which matches. The common error is to copy the sign of $b$ into the bracket unchanged.

Re-check: expand your completed-square form and confirm it gives back the original expression, term by term.

Progress check

Return to the chapter problem. The arch is modelled by $y = 4 - \frac{x^2}{4}$.

1. Write the model in completed-square form, $a(x + p)^2 + q$.
2. State the greatest height of the arch and where it occurs.
3. Solve $4 - \frac{x^2}{4} = 0$ to find where the arch meets the road.
4. The equation $4 - \frac{x^2}{4} = 5$ asks for a point higher than the arch. Work out the discriminant of $-\frac{x^2}{4} - 1 = 0$ and explain what its sign says about the request.

You should now be able to

• write a quadratic in completed-square form, for $a = 1$ and for $a \neq 1$;
• read the vertex, the line of symmetry and the minimum or maximum value;
• work out the discriminant and use its sign to count the real roots;
• connect the number of crossings of the $x$-axis with the discriminant;
• expand a completed square to check it against the original expression.