The Quadratic Formula and the Nature of Roots: Quick Revision

A brief note on The Quadratic Formula and the Nature of Roots. Read it before a test. Try each question at the end before you open its answer.

Completing the square

• $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$.
• Move the number term, add $\left(\frac{b}{2}\right)^2$ to both sides, take the square root with $\pm$, and solve.
• With $ax^2$, divide every term by $a$ first.
• If the right side is negative, there are no real roots.

The quadratic formula

• For $ax^2 + bx + c = 0$, with $a \neq 0$:

• Read $a$, $b$ and $c$ with their signs. Work out $b^2 - 4ac$ first.
• Check each root by putting it back, and keep only the roots that make sense in a problem.

The discriminant

• $D = b^2 - 4ac$.
• $D > 0$: two distinct real roots.
• $D = 0$: two equal real roots, $x = -\dfrac{b}{2a}$.
• $D < 0$: no real roots.
• To make the roots equal, set $D = 0$ and solve for the unknown number.

A worked example

Solve $4x^2 - 4x - 1 = 0$.

1. The numbers. $a = 4$, $b = -4$, $c = -1$.
2. The discriminant. $D = 16 + 16 = 32$. It is positive, so there are two distinct roots.
3. The formula. $x = \dfrac{4 \pm \sqrt{32}}{8} = \dfrac{4 \pm 4\sqrt{2}}{8} = \dfrac{1 \pm \sqrt{2}}{2}$.
4. The numbers. $x \approx 1.207$ or $x \approx -0.207$.

Practice with new numbers

A program to try

The program works out the discriminant of four equations and solves each one that has real roots.

Mistakes to avoid

• Adding $b^2$ when completing the square. Add the square of half of $b$.
• Losing a sign. With $b = -4$, $-b = 4$, and $b^2 = 16$.
• Dividing only the root by $2a$. The whole top is divided by $2a$.
• Saying "no roots" when $D = 0$. There is one root, counted twice.

Check yourself

1. Which number completes the square for $x^2 + 18x$?

2. Solve $x^2 - 6x + 7 = 0$.

3. What kind of roots does $3x^2 + 2x + 1 = 0$ have?

4. For which $k$ does $x^2 - 8x + k = 0$ have two equal roots?

5. Why must $a$ not be $0$ in the quadratic formula?