Quadratic Equations of Powers

Maths: Edexcel A Level Pure Maths, topic 2.2.10

What are Quadratic Equations of Powers?

Quadratic equations of powers are equations of the form $ap^2 + bp+c$ where $a,b,c$ are constants and $p$ is a power of $x$ .

What are quadratic equations of powers?

What are quadratic equations of powers?

Quadratic equations of powers will always contain at least two terms:
A term with the highest power of $x$ : $x^{2n}$ .
The term with half the power of the highest power of $x$ : $x^n$ .
Quadratic equations of powers can also include a constant term.

How do you solve quadratic equations of powers?

How do you solve quadratic equations of powers?

In order to find the roots of a quadratic equation of a power, rewrite it so that it looks like a regular quadratic equation:
$ax^{2n} + bx^n + c = 0\rightarrow a(x^n)^2 + b(x^n) +c = 0$
We can then factorise this if it were a regular quadratic equation.

Example

Example

What is the solution to the equation $f(x) = x^6 - 16x^3 + 64= 0$ ?

Inspection

Inspection

We can make this equation look more like a quadratic by grouping the factors of $x^3$ in the first two terms:
$(x^3)^2 - 16(x^3) + 64 = 0$

Factorise

Factorise

We can factorise this equation as if it were a simple quadratic:
$(x^3)^2 - 16(x^3) + 64 \equiv (x^3 - 8)^2$
$(x^3 - 8)^2 = 0$

Solve for $$x^3$$

Solve for $x^3$

By taking the square root of both sides, we get:
$x^3 - 8 = 0$
Rearranging this we get:
$x^3 = 8$
Finally, we can take the cube root of both sides, which gives:
$x = 2$

Full solution

Full solution

Test yourself on Quadratic Equations of Powers

Which of the following equations would be used to find the roots of the function $f(x) = x^3 + x^{\frac{3}{2}} - 3$ ?
Which of the following equations would be used to find the roots of the function $f(x) = 6x^{12} + x^{6} - 7$ ?

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