KnowledgeBoat Logo
|

Mathematics

Solve the following equation using the formula :

4x−3=52x+3\dfrac{4}{x} - 3 = \dfrac{5}{2x + 3}

Quadratic Equations

11 Likes

Answer

Given,

⇒4x−3=52x+3⇒4−3xx=52x+3⇒(4−3x)(2x+3)=5x⇒8x+12−6x2−9x=5x⇒−6x2−x+12−5x=0⇒−6x2−6x+12=0⇒−6(x2+x−2)=0⇒x2+x−2=0\Rightarrow \dfrac{4}{x} - 3 = \dfrac{5}{2x + 3} \\[1em] \Rightarrow \dfrac{4 - 3x}{x} = \dfrac{5}{2x + 3} \\[1em] \Rightarrow (4 - 3x)(2x + 3) = 5x \\[1em] \Rightarrow 8x + 12 - 6x^2 - 9x = 5x \\[1em] \Rightarrow -6x^2 - x + 12 - 5x = 0 \\[1em] \Rightarrow -6x^2 - 6x + 12 = 0 \\[1em] \Rightarrow -6(x^2 + x - 2) = 0 \\[1em] \Rightarrow x^2 + x - 2 = 0

Comparing x2 + x - 2 = 0 with ax2 + bx + c = 0 we get,

a = 1, b = 1, c = -2.

We know that,

x = −b±b2−4ac2a\dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substituting values we get,

⇒x=−(1)±(1)2−4(1)(−2)2(1)=−1±1+82=−1±92=−1±32=−1+32 or −1−32=22 or −42=1 or −2.\Rightarrow x = \dfrac{-(1) \pm \sqrt{(1)^2 - 4(1)(-2)}}{2(1)} \\[1em] = \dfrac{-1 \pm \sqrt{1 + 8}}{2} \\[1em] = \dfrac{-1 \pm \sqrt{9}}{2} \\[1em] = \dfrac{-1 \pm 3}{2} \\[1em] = \dfrac{-1 + 3}{2} \text{ or } \dfrac{-1 - 3}{2} \\[1em] = \dfrac{2}{2} \text{ or } \dfrac{-4}{2} \\[1em] = 1 \text{ or } -2.

Hence, x = 1, -2.

Answered By

4 Likes

Related Questions