1.3 Complex Quadratics Worksheet

Quick Notes & Formulae

Essential strategies for identifying and determining hidden quadratics.

Substitution

Replace terms like

x^{4}

\sqrt{x}

, or

a^{2 x}

with

u

to form a quadratic

a u^{2} + b u + c = 0

Validity Check

Ensure

u

values satisfy conditions, e.g.,

u = \sqrt{x} \geq 0

u = a^{x} > 0

Solving Intersections

To find where line meets curve, substitute line equation into curve equation and solve.

Surd Equations

Isolate the square root term before squaring both sides to remove radicals.

Midpoint Formula

M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})

Distance Formula

d = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}}

Past Paper Style Questions

Question 1: Higher Powers Substitutions

Find the real values of

x

for:
(a)

x^{8} - 20 x^{4} + 64 = 0

(b)

x^{6} - 4 x^{3} - 45 = 0

Question 2: Further Polynomials

Find the real values of

x

for:
(a)

2 x^{4} - 9 x^{2} + 4 = 0

(b)

x^{4} - 6 x^{2} + 8 x - 3 = 0

Question 3: Surd Substitution

Solve for real

x

:
(a)

4 x - 9 \sqrt{x} + 2 = 0

Question 4: Fractions with Roots

Solve:
(a)

\frac{5}{\sqrt{x}} + \sqrt{x} = 6

(b)

\frac{12}{\sqrt{x}} - \sqrt{x} = 1

Question 5: Double Radical Equation

Solve for real

x

\sqrt{x + 3} + \sqrt{x - 1} = 4

Question 6: Unknown Coefficients (Two Points)

The graph shows

y = a x + b \sqrt{x} + c

for

x \geq 0

. It crosses the x-axis at

(4, 0)

and

(36, 0)

, and meets the y-axis at

(0, 10)

. Find

a, b, c

Question 7: Unknown Coefficients (Three Points)

The graph shows

y = a x + b \sqrt{x} + c

. It crosses the x-axis at

(9, 0)

and

(25, 0)

, and passes through

(16, 8)

. Find

a, b, c

Question 8: Intersection and Midpoint

Curve

y = 3 \sqrt{x} + 2

and line

y = x + 4

intersect at

A

and

B

. Find coordinates of

A

B

and the midpoint of

A B

Question 9: Line Intersection

Curve

y = 5 \sqrt{x} - x

and line

y = 2 x + 1

intersect at

A, B

. Find coordinates.

Question 10: Mixed Substitution

Solve for

x

x - 4 \sqrt{x} = 21