MPM2D The Quadratic Formula: Notes and Practice

A rectangular garden bed is 1 m longer than it is wide, and it must have an area of exactly 4 m². If the width is $x$ metres, the length is $x + 1$ and the area equation is $x (x + 1) = 4$ , or $x^{2} + x - 4 = 0$ . Try to factor it: you need two integers with product $- 4$ and sum $1$ . The candidates are $1$ and $- 4$ (sum $- 3$ ), $- 1$ and $4$ (sum $3$ ), and $2$ and $- 2$ (sum $0$ ). None works. Yet a garden that is 1 m wide has area $1 \times 2 = 2$ m² and one that is 2 m wide has area $2 \times 3 = 6$ m², so some width between 1 m and 2 m gives exactly 4 m². The root exists; it just is not a whole number or a simple fraction. You need a method that does not depend on lucky factors. That method is the quadratic formula, and it comes from completing the square, which you used in the last unit to find vertex form.

A rectangular raised garden bed with brightly painted wooden sides on a lawn in a community garden — PhotoBuilding a rectangular bed with an exact area often leads to a quadratic equation that will not factor, which is when the quadratic formula is needed.Bart Everson from New Orleans, Louisiana, USA, CC BY 2.0, via Wikimedia Commons

Where the formula comes from

The plan is to complete the square on the general equation $a x^{2} + b x + c = 0$ , where the letters stand for any numbers. Because the letters make the algebra hard to follow, the table runs a numerical example beside the general case. Read across each row: the same step is done to both. You do not need to reproduce the general case yourself, but you should be able to explain what each step does by pointing at the numerical example.

Step	Example: 2x² + 12x + 4 = 0	General: ax² + bx + c = 0
1. Start in standard form	$2 x^{2} + 12 x + 4 = 0$	$a x^{2} + b x + c = 0$
2. Divide every term by $a$	$x^{2} + 6 x + 2 = 0$	$x^{2} + \frac{b}{a} x + \frac{c}{a} = 0$
3. Move the constant to the right side	$x^{2} + 6 x = - 2$	$x^{2} + \frac{b}{a} x = - \frac{c}{a}$
4. Add the square of half the coefficient of $x$ to both sides	$x^{2} + 6 x + 9 = - 2 + 9$	$x^{2} + \frac{b}{a} x + \frac{b ^{2}}{4 a ^{2}} = \frac{b ^{2}}{4 a ^{2}} - \frac{c}{a}$
5. Write the left side as a square and combine the right side	$(x + 3)^{2} = 7$	$(x + \frac{b}{2 a})^{2} = \frac{b ^{2} - 4 a c}{4 a ^{2}}$
6. Take the square root of both sides, with ±	$x + 3 = \pm 7$	$x + \frac{b}{2 a} = \pm \frac{b ^{2} - 4 a c}{2 a}$
7. Subtract to isolate $x$	$x = - 3 \pm 7$	$x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$

Completing the square on a numerical example and on the general equation, step by step

A few steps need a word of explanation. In step 4, half of $\frac{b}{a}$ is $\frac{b}{2 a}$ , and its square is $\frac{b ^{2}}{4 a ^{2}}$ ; in the example, half of 6 is 3 and $3^{2} = 9$ . In step 5 the right side is combined over the common denominator $4 a^{2}$ : $\frac{b ^{2}}{4 a ^{2}} - \frac{4 a c}{4 a ^{2}} = \frac{b ^{2} - 4 a c}{4 a ^{2}} .$ In step 6 the $\pm$ is there because both a positive and a negative number square to the same value: $(7)^{2} = 7$ and $(- 7)^{2} = 7$ . The square root of the denominator $4 a^{2}$ is written $2 a$ ; if $a$ is negative it is really $- 2 a$ , but the $\pm$ already covers both signs, so the two roots come out the same. Finally, check that the formula agrees with the example: with $a = 2$ , $b = 12$ , $c = 4$ it gives $x = \frac{- 12 \pm 144 - 32}{4} = \frac{- 12 \pm 112}{4}$ , which is about $- 0.354$ or $- 5.646$ . The completed square gave $- 3 \pm 7$ , which is about $- 3 \pm 2.646$ , the same two numbers.

a x^{2} + b x + c = 0, a \neq = 0 ⟹ x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

The quadratic formula. The + sign gives one root and the − sign gives the other.

Fig 1Top: the quadratic formula with each part labelled: −b, the ± sign, the expression b² − 4ac under the square root, and 2a under the whole numerator. Bottom: the same formula split into −b over 2a plus or minus a distance, shown on a number line for x² + x − 4 = 0, where the roots about −2.56 and 1.56 sit 2.06 on either side of −0.5.

The figure shows a useful way to read the formula. Split the single fraction into two parts: $x = - \frac{b}{2 a} \pm \frac{b ^{2} - 4 a c}{2 a}$ . The first part, $- \frac{b}{2 a}$ , is the same for both roots: it is the $x$ -coordinate of the vertex, on the axis of symmetry. The second part is how far each root sits from that axis, once to the right and once to the left. That is why, whenever a parabola has two zeros, they are the same distance from its axis of symmetry.

VideoCompleting the square on the general equation ax squared plus bx plus c equals zero produces the quadratic formula step by step.Watch on YouTube

Using the formula

Write the equation in standard form, $a x^{2} + b x + c = 0$ .
Read off $a$ , $b$ and $c$ , including their signs. In $3 x^{2} - 5 x - 1 = 0$ , $a = 3$ , $b = - 5$ and $c = - 1$ .
Work out $b^{2} - 4 a c$ first, on its own. It is the most error-prone part, and its value tells you a lot (next lesson).
Substitute into the formula, with brackets around negative values.
Write the two roots, one with $+$ and one with $-$ . Leave them exact, or round only at the very end.

Back to the garden, $x^{2} + x - 4 = 0$ . Here $a = 1$ , $b = 1$ and $c = - 4$ , so $b^{2} - 4 a c = 1^{2} - 4 (1) (- 4) = 1 + 16 = 17 .$ Then $x = \frac{- 1 \pm 17}{2}$ . These are the exact roots. Since $17 \approx 4.1231$ , the roots are $x \approx \frac{- 1 + 4.1231}{2} \approx 1.56$ and $x \approx \frac{- 1 - 4.1231}{2} \approx - 2.56$ , rounded to two decimal places. A width cannot be negative, so the garden is about 1.56 m wide and 2.56 m long. Check: $1.56 \times 2.56 \approx 3.99$ , which rounds to 4. To verify graphically, graph $y = x^{2} + x - 4$ with graphing technology: the parabola crosses the $x$ -axis near $- 2.56$ and $1.56$ .

Exact roots and rounded roots

An exact root keeps the square root, as in

\frac{- 1 + 17}{2}

. A rounded (approximate) root is a decimal, as in

1.56

. Give exact roots when a question asks for them. Otherwise, keep the full calculator value until the final step and round only the answer. Rounding

17

4

partway through gives

\frac{- 1 + 4}{2} = 1.5

, which is wrong in the second decimal place.

The formula also works on equations that factor. For $x^{2} + 10 x + 16 = 0$ : $b^{2} - 4 a c = 100 - 64 = 36$ and $36 = 6$ , so $x = \frac{- 10 \pm 6}{2}$ , which gives $\frac{- 4}{2} = - 2$ and $\frac{- 16}{2} = - 8$ , the same roots as factoring. Notice that 36 is a perfect square. When $a$ , $b$ and $c$ are integers and $b^{2} - 4 a c$ is a perfect square, the roots are rational and the equation factors with integers. When it is not a perfect square, like 17, the roots are irrational, and factoring with integers was never going to work.

Method	Best when	What it gives
Factoring	Small integer coefficients and a factorable expression, such as $x^{2} - 3 x - 10 = 0$	Exact roots, quickly
Quadratic formula	Any quadratic equation, especially one that does not factor	Exact roots (possibly with a square root) or decimals
Graphing technology	Checking answers, or when only approximate roots are needed	Approximate roots read from the x-intercepts

Three ways to solve a quadratic equation

Brackets around negatives

b = - 5

, then

- b = 5

and

b^{2} = (- 5)^{2} = 25

. Typing

- 5^{2}

into a calculator gives

- 25

, because the calculator squares 5 first and then applies the minus sign. Type

(- 5)^{2}

, with brackets. Also remember that

- 4 a c

becomes positive when

a

and

c

have opposite signs: with

a = 3

and

c = - 1

- 4 (3) (- 1) = + 12

The Quadratic Formula

Key things to remember

Where the formula comes from

Using the formula

Worked examples

Where marks get lost

Flashcards 7

Practice