Educational

Solve the Following Quadratic Equation by Factorization Question: Solve the quadratic equation: \[ \frac{16}{x}-1=\frac{15}{x+1}, \qquad x\ne 0,-1 \] Solution Given: \[ \frac{16}{x}-1=\frac{15}{x+1} \] Multiplying both sides by \(x(x+1)\), we get: \[ 16(x+1)-x(x+1)=15x \] \[ 16x+16-x^2-x=15x \] \[ -x^2+15x+16=15x \] \[ -x^2+16=0 \] \[ x^2-16=0 \] Using the identity \(a^2-b^2=(a-b)(a+b)\): \[ (x-4)(x+4)=0 \] Therefore, \[ x-4=0 […]

Solve the Quadratic Equation by Factorization: 2x² + ax − a² = 0 Question: Solve the following quadratic equation by factorization: $$ 2x^2+ax-a^2=0 $$ Solution $$ 2x^2+2ax-ax-a^2=0 $$ $$ 2x(x+a)-a(x+a)=0 $$ $$ (x+a)(2x-a)=0 $$ Either $$ x+a=0 $$ $$ x=-a $$ or $$ 2x-a=0 $$ $$ x=\frac{a}{2} $$ Hence, $$ \boxed{x=-a,\ \frac{a}{2}} $$ Next Question

Solve the Quadratic Equation by Factorization: ax² + (4a² − 3b)x − 12ab = 0 Question: Solve the following quadratic equation by factorization: $$ ax^2+(4a^2-3b)x-12ab=0 $$ Solution $$ ax^2+4a^2x-3bx-12ab=0 $$ $$ ax(x+4a)-3b(x+4a)=0 $$ $$ (x+4a)(ax-3b)=0 $$ Either $$ x+4a=0 $$ $$ x=-4a $$ or $$ ax-3b=0 $$ $$ x=\frac{3b}{a} $$ Hence, $$ \boxed{x=-4a,\ \frac{3b}{a}} $$

Solve the Quadratic Equation by Factorization: 4x² + 4bx − (a² − b²) = 0 Question: Solve the following quadratic equation by factorization: $$ 4x^2+4bx-(a^2-b^2)=0 $$ Solution $$ 4x^2+4bx+b^2-a^2=0 $$ $$ (2x+b)^2-a^2=0 $$ $$ (2x+b-a)(2x+b+a)=0 $$ Either $$ 2x+b-a=0 $$ $$ x=\frac{a-b}{2} $$ or $$ 2x+b+a=0 $$ $$ x=-\frac{a+b}{2} $$ Hence, $$ \boxed{x=\frac{a-b}{2},\ -\frac{a+b}{2}} $$

Solve the Quadratic Equation by Factorization: 9x² − 6b²x − (a⁴ − b⁴) = 0 Question: Solve the following quadratic equation by factorization: $$ 9x^2-6b^2x-(a^4-b^4)=0 $$ Solution $$ 9x^2-6b^2x-a^4+b^4=0 $$ $$ 9x^2-6b^2x+b^4-a^4=0 $$ $$ (3x-b^2)^2-a^4=0 $$ $$ (3x-b^2-a^2)(3x-b^2+a^2)=0 $$ Either $$ 3x-b^2-a^2=0 $$ $$ x=\frac{a^2+b^2}{3} $$ or $$ 3x-b^2+a^2=0 $$ $$ x=\frac{b^2-a^2}{3} $$ Hence, $$

Solve the Quadratic Equation by Factorization: a²x² − 3abx + 2b² = 0 Question: Solve the following quadratic equation by factorization: $$ a^2x^2-3abx+2b^2=0 $$ Solution $$ a^2x^2-2abx-abx+2b^2=0 $$ $$ ax(ax-2b)-b(ax-2b)=0 $$ $$ (ax-2b)(ax-b)=0 $$ Either $$ ax-2b=0 $$ $$ x=\frac{2b}{a} $$ or $$ ax-b=0 $$ $$ x=\frac{b}{a} $$ Hence, $$ \boxed{x=\frac{2b}{a},\ \frac{b}{a}} $$ Next Question

Solve the Quadratic Equation by Factorization: 1/(x − 3) + 2/(x − 2) = 8/x Question: $$ \frac{1}{x-3}+\frac{2}{x-2}=\frac{8}{x}, \quad x\ne0,2,3 $$ Solution $$ \frac{x-2+2(x-3)}{(x-3)(x-2)}=\frac{8}{x} $$ $$ \frac{3x-8}{(x-3)(x-2)}=\frac{8}{x} $$ $$ x(3x-8)=8(x-3)(x-2) $$ $$ 3x^2-8x=8x^2-40x+48 $$ $$ 5x^2-32x+48=0 $$ $$ 5x^2-20x-12x+48=0 $$ $$ 5x(x-4)-12(x-4)=0 $$ $$ (x-4)(5x-12)=0 $$ Either $$ x-4=0 $$ $$ x=4 $$ or $$

Solve the Quadratic Equation by Factorization: 1/(x + 4) − 1/(x − 7) = 11/30 Question: $$ \frac{1}{x+4}-\frac{1}{x-7}=\frac{11}{30}, \quad x\ne -4,7 $$ Solution $$ \frac{(x-7)-(x+4)}{(x+4)(x-7)}=\frac{11}{30} $$ $$ \frac{-11}{(x+4)(x-7)}=\frac{11}{30} $$ $$ (x+4)(x-7)=-30 $$ $$ x^2-3x-28=-30 $$ $$ x^2-3x+2=0 $$ $$ x^2-2x-x+2=0 $$ $$ x(x-2)-1(x-2)=0 $$ $$ (x-2)(x-1)=0 $$ Either $$ x-2=0 $$ $$ x=2 $$ or

Solve the Quadratic Equation by Factorization: x − 1/x = 3 Question: $$ x-\frac{1}{x}=3, \quad x\ne0 $$ Solution $$ x^2-1=3x $$ $$ x^2-3x-1=0 $$ $$ x^2-\frac{3+\sqrt{13}}{2}x+\frac{3+\sqrt{13}}{2}x-1=0 $$ $$ \left(x-\frac{3+\sqrt{13}}{2}\right)\left(x-\frac{3-\sqrt{13}}{2}\right)=0 $$ Therefore, $$ x=\frac{3+\sqrt{13}}{2} \quad \text{or} \quad x=\frac{3-\sqrt{13}}{2} $$ Hence, $$ \boxed{x=\frac{3+\sqrt{13}}{2},\ \frac{3-\sqrt{13}}{2}} $$ Next Question / Full Exercise

Solve the Quadratic Equation by Factorization: 1/x − 1/(x − 2) = 3 Question: $$ \frac{1}{x}-\frac{1}{x-2}=3, \quad x\ne0,2 $$ Solution $$ \frac{x-2-x}{x(x-2)}=3 $$ $$ \frac{-2}{x(x-2)}=3 $$ $$ -2=3x(x-2) $$ $$ 3x^2-6x+2=0 $$ $$ 3x^2-3x-3x+2=0 $$ $$ 3x(x-1)-2(x-1)=0 $$ $$ (3x-2)(x-1)=0 $$ Either $$ 3x-2=0 $$ $$ x=\frac{2}{3} $$ or $$ x-1=0 $$ $$ x=1 $$