Q.Find values of $x$, if

Compute each $2\times2$ determinant with the rule $ad-bc$, set the two sides equal, and solve for $x$.

(i) $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = 2-20 = -18$ and $\begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix} = 2x^2-24$.

Equate: $2x^2-24=-18 \Rightarrow 2x^2=6 \Rightarrow x^2=3 \Rightarrow x=\pm\sqrt{3}$. …

A $2\times2$ determinant $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is just the number $ad-bc$. So an equation between two determinants says two numbers are equal — evaluate both, then solve for $x$.

(i)

Left: $\begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = (2)(1)-(4)(5) = 2-20 = -18.$

Right: $\begin{vmatrix} 2x & 4 \\ 6 & x \end{vmatrix} = (2x)(x)-(4)(6) = 2x^2-24.$

Equate: $2x^2-24=-18 \Rightarrow 2x^2=6 \Rightarrow x^2=3 \Rightarrow x=\pm\sqrt{3}.$ …

Method: Solving "Find $x$" Equations Between Two Determinants

Applies whenever two $2\times2$ (or larger) determinants are set equal and one contains an unknown.

Steps

Step 1: Evaluate the side with no unknown (or the simpler side) to a single number

Use $ad-bc$ to reduce it to a pure numeric value.

Step 2: Evaluate the side containing $x$ in terms of $x$

Apply $ad-bc$ again; if $x$ appears in more than one entry, the result will generally be a polynomial in $x$ (often quadratic, since $x$ may be squared).

Step 3: Equate the two expressions and solve …

Common Mistakes

Mistake 1: Matching individual matrix entries instead of equating the two determinant VALUES

Why it's wrong: the problem states that the two determinants (numbers) are equal, not that the two matrices are equal entry by entry — trying to match entries directly is solving the wrong condition and generally has no valid solution. Correct approach: evaluate both determinants to numbers/expressions first, then set those two results equal.

Mistake 2: Reporting only one root when the equation in $x$ turns out to be quadratic …