Quadratic equations with non-real roots | Year 11

01 / LEARN

Understand the idea.

A negative discriminant is not the end

For a real-coefficient quadratic $a z^{2} + b z + c = 0$ with $a \neq 0$ , the discriminant is $Δ = b^{2} - 4 a c$ . When $Δ < 0$ , there are no real roots, but the complex quadratic formula still gives two non-real roots.

Use $\sqrt{Δ} = i \sqrt{- Δ}$ in the numerator, keep both signs, and simplify the real and imaginary components separately. The denominator $2 a$ divides the entire numerator, including the imaginary term.

Completing the square and checking

Completing the square may be quicker when the coefficients are simple. A form ${(z - h)}^{2} = - k$ with $k > 0$ yields $z - h = \pm i \sqrt{k}$ , then $z = h \pm i \sqrt{k}$ . Keep both signs when reversing the square.

Check roots in the original equation or the equivalent completed-square form. Their real and imaginary contributions must cancel exactly. Exact radicals and fractions make the cancellation clearer than rounded decimal approximations.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Complete the square

Solve $z^{2} - 6 z + 13 = 0$ .

Worked solution

1
Complete the square.
$z^{2} - 6 z + 13 = {(z - 3)}^{2} + 4$
The square ${(z - 3)}^{2}$ accounts for the quadratic and linear terms and contributes nine to the constant.
2
Rearrange the equation.
${(z - 3)}^{2} = - 4$
Subtract four after setting the completed-square expression equal to zero.
3
Take both square roots.
$z - 3 = \pm 2 i ⟹ z = 3 \pm 2 i$
Both imaginary signs square to negative four.
4
Check in the equivalent equation.
${(\pm 2 i)}^{2} + 4 = - 4 + 4 = 0$
Both roots make the completed-square expression zero.

Answer

$z = 3 \pm 2 i$ .

EXAMPLE 02

A non-monic quadratic

Solve $2 z^{2} + 4 z + 10 = 0$ .

Worked solution

1
Divide by the nonzero leading coefficient.
$2 z^{2} + 4 z + 10 = 0 ⟹ z^{2} + 2 z + 5 = 0$
Dividing every term by two preserves the roots.
2
Calculate the discriminant.
$Δ = 2^{2} - 4 (1) (5) = - 16$
The negative value requires imaginary square roots.
3
Apply and simplify the quadratic formula.
$z = \frac{- 2 \pm 4 i}{2} = - 1 \pm 2 i$
The principal square root of negative sixteen is four i, and the formula supplies both signs.
4
Check both roots after completing the square.
${(z + 1)}^{2} + 4 = {(\pm 2 i)}^{2} + 4 = 0$
The normalised quadratic is equivalent to this completed-square equation.

Answer

$z = - 1 \pm 2 i$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

Solve $z^{2} + 2 z + 5 = 0$ .

Show answer & reasoning

$z = - 1 \pm 2 i$ .

Complete the square: ${(z + 1)}^{2} = - 4$ . Both square roots give $z + 1 = \pm 2 i$ , so $z = - 1 \pm 2 i$ .

Keep practising

6 worksheet questions · 16 marks

Open worksheet Worked solutions

Read the questions online

Solve $z^{2} + 25 = 0$ .
2 marks
Solve $z^{2} - 4 z + 8 = 0$ .
3 marks
Solve $3 z^{2} + 12 = 0$ .
2 marks
Solve $z^{2} + 6 z + 11 = 0$ exactly.
3 marks
Check directly that $1 + 3 i$ solves $z^{2} - 2 z + 10 = 0$ .
3 marks
For real $k$ , determine when $z^{2} + 4 z + k = 0$ has non-real roots.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Solve $z^{2} + 25 = 0$ .
2 marks
$z^{2} = - 25$ , whose two roots are $\pm 5 i$ .
$z = \pm 5 i$ .
Solve $z^{2} - 4 z + 8 = 0$ .
3 marks
Complete the square: ${(z - 2)}^{2} = - 4$ .
$z = 2 \pm 2 i$ .
Solve $3 z^{2} + 12 = 0$ .
2 marks
Divide by $3$ to obtain $z^{2} = - 4$ .
$z = \pm 2 i$ .
Solve $z^{2} + 6 z + 11 = 0$ exactly.
3 marks
${(z + 3)}^{2} = - 2$ .
Take both roots and subtract $3$ .
$z = - 3 \pm i \sqrt{2}$ .
Check directly that $1 + 3 i$ solves $z^{2} - 2 z + 10 = 0$ .
3 marks
${(1 + 3 i)}^{2} = - 8 + 6 i$ .
$(- 8 + 6 i) - 2 (1 + 3 i) + 10 = 0 + 0 i$ .
The substitution gives zero.
For real $k$ , determine when $z^{2} + 4 z + k = 0$ has non-real roots.
3 marks
The discriminant is $16 - 4 k$ .
It is negative exactly when $16 - 4 k < 0$ , or $k > 4$ .
$k > 4$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics, real and complex numbers: Cartesian arithmetic, conjugates, quadratic roots and Argand geometry. Polar form and roots of unity are developed in Year 12.

Official curriculum source ↗

In this lesson

Understand the idea.

A negative discriminant is not the end

Completing the square and checking

See the method in action.

Complete the square

Complete the square.

Rearrange the equation.

Take both square roots.

Check in the equivalent equation.

A non-monic quadratic

Divide by the nonzero leading coefficient.

Calculate the discriminant.

Apply and simplify the quadratic formula.

Check both roots after completing the square.

Try it yourself.

Check your understanding

Keep practising