how to solve transforming rational algebraic equations to quadratic equations questions
Answer (1)
To solve the equation 2x/3 + x+1/x = 9 we can follow these steps to transform it into a quadratic equation:Step 1: Find a common denominatorThe denominators in the equation are 3 and x. The least common denominator (LCD) is 3x. Multiply both sides of the equation by 3x to eliminate the fractions.3x ({2x/3} + {x+1}{x}) = (3x)9 Distribute 3x to each term:[tex]\[3x \times \frac{2x}{3} + 3x \times \frac{x+1}{x} = 27x\][/tex]Simplifying each term:[tex] \( 3x \times \frac{2x}{3} = 2x^2 \) \\ \( 3x \times \frac{x+1}{x} = 3(x + 1) = 3x + 3 \)[/tex]The equation now becomes:[tex]\[2x^2 + 3x + 3 = 27x\][/tex]Step 2: Move all terms to one sideSubtract 27x from both sides of the equation to set it to zero:[tex]\[2x^2 + 3x + 3 - 27x = 0\][/tex]Simplify:[tex]\[2x^2 - 24x + 3 = 0\][/tex]Step 3: Solve the quadratic equationNow that we have the quadratic equation 2x² - 24x + 3 = 0, we can solve it using either the quadratic formula or factoring (if possible).Quadratic Formula:The quadratic formula is given by:[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]For the equation 2x² - 24x + 3 = 0,the coefficients are:[tex]\( a = 2 \) \\ \( b = -24 \) \\ \( c = 3 \)[/tex]Substituting these into the quadratic formula:[tex]\[x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4(2)(3)}}{2(2)}\][/tex][tex]\[x = \frac{24 \pm \sqrt{576 - 24}}{4}\][/tex][tex]\[x = \frac{24 \pm \sqrt{552}}{4}\][/tex][tex]\[x = \frac{24 \pm 2\sqrt{138}}{4}\][/tex][tex]\[x = \frac{12 \pm \sqrt{138}}{2}\][/tex]Thus, the solutions are:[tex]\[x = \frac{12 + \sqrt{138}}{2} \quad \text{or} \quad x = \frac{12 - \sqrt{138}}{2}\][/tex]This gives you the two possible values for x after solving the quadratic equation.CARRY ON LEARNING!
