A root of the equation Δ$$=0x$$−ax−$$bx+a0x$$−$$cx+bx+c0=0$$ is

Question

A root of the equation Δ$$=0x$$−ax−$$bx+a0x$$−$$cx+bx+c0=0$$ is

Infinity Learn · Accepted Answer

When we substitute x $$=$$ $$0$$, ∆ becomes      $$0$$−a−$$ba0$$−$$cbc0which$$ is equal to $$0$$ as ∆ is skew symmetric determinant of odd order.Alternative Solution Evaluating ∆ along $$R1$$, we get−(x$$−$$a)x+ax$$−$$cx+b0+(x$$−$$b)x+a0x+bx+c=0$$⇒$$(x$$−$$a)(x+b)(x$$−$$c)+(x$$−$$b)(x+a)(x+c)=0$$⇒$$x3+($$−$$a+b$$−$$c)x2+($$−$$ab+ac$$−$$bc)x+abc+x3+(a$$−$$b+c)x2+($$−$$ab+ac$$−$$bc)−$$abc=0$$⇒$$2x3$$−$$2(ab$$−$$ac+bc)x=0$$⇒$$2xx2$$−$$(ab$$−$$ac+bc)=0$$ ⇒$$x=0$$ or $$x=$$±ab−$$ac+bc$$∴ one of the roots of the equation is $$0$$.

A root of the equation $Δ = |\begin{array}{ccc} 0 & x - a & x - b \\ x + a & 0 & x - c \\ x + b & x + c & 0 \end{array}| = 0$ is

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A root of the equation Δ=0x−ax−bx+a0x−cx+bx+c0=0 is

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A root of the equation $Δ = |\begin{array}{ccc} 0 & x - a & x - b \\ x + a & 0 & x - c \\ x + b & x + c & 0 \end{array}| = 0$ is