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Equations that are quadratic in a function of x · Fathom

Equations that are quadratic in a function of x

5 minAS Level1.1.5

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can spot an equation that is quadratic in some function of $x$ (like $x^2$ , $\sqrt{x}$ or $\tan x$ ), solve it with a substitution, and convert back to $x$ , rejecting any impossible values.

Spot the disguise

A quadratic doesn't have to be in $x$ . A quadratic in $u$ looks like $au^2 + bu + c = 0$ , where $u$ can stand for anything.

The test: if one term contains the square of what another term contains, you've got a quadratic in that thing.

For example, $x^4 = (x^2)^2$ . So $x^4 - 13x^2 + 36$ has " $x^2$ " and " $x^2$ squared". It's a quadratic in $x^2$ .

Equation	Let	It becomes
$x^4 - 13x^2 + 36 = 0$	$u = x^2$	$u^2 - 13u + 36 = 0$
$x - \sqrt{x} - 6 = 0$	$u = \sqrt{x}$	$u^2 - u - 6 = 0$
$\dfrac{2}{x^2} - \dfrac{3}{x} + 1 = 0$	$u = \dfrac{1}{x}$	$2u^2 - 3u + 1 = 0$
$x^6 - 7x^3 - 8 = 0$	$u = x^3$	$u^2 - 7u - 8 = 0$
$2\tan^2 x + \tan x - 3 = 0$	$u = \tan x$	$2u^2 + u - 3 = 0$

The tell: powers in the ratio 2 : 1.

$x^4$ and $x^2$ : the power 4 is twice 2.
$x$ and $\sqrt{x}$ : a power of $\tfrac12$ means a square root, so $\sqrt{x} = x^{\frac12}$ . The power 1 is twice $\tfrac12$ .
$\tan^2 x$ and $\tan x$ : here $\tan^2 x$ means $(\tan x)^2$ .

Worked example: A quartic in disguise

Solve $x^4 - 13x^2 + 36 = 0$ .

Have a go first?

Quick check

Multiple choiceQuestion 1 · Apply

CalculateQuestion 2 · Apply

Answer the 2 questions above to continue.