To solve the quadratic equation, we need to rearrange it into the standard form ax² + bx + c = 0. We do this by moving all terms to one side of the equation and combining like terms.

Find the roots of the equation $\frac{1}{2x-3} + \frac{1}{x-5} = 1$ $x \neq \frac{3}{2}, 5$ (roots, equation, fractions, domain, quadratic)

Answer: The roots of the equation are $x = \frac{8 + 3\sqrt{2}}{2}$ and $x = \frac{8 - 3\sqrt{2}}{2}$ .

Step-by-step solution

Step 1: Combine fractions on the left side

To solve this equation, we first need to combine the fractions on the left-hand side. We do this by finding a common denominator, which is the product of the two denominators: $(2x-3)(x-5)$ . Then, we cross-multiply the numerators with the opposite denominators and add them.

Step 2: Simplify the numerator and cross-multiply

Now, we simplify the numerator on the left side by combining like terms. Then, we multiply both sides of the equation by the denominator $(2x-3)(x-5)$ to eliminate the fraction, moving it to the right side.

Step 3: Expand the right side

We expand the product of the two binomials on the right side of the equation using the distributive property (FOIL method). This will result in a quadratic expression.

Step 4: Rearrange into standard quadratic form

To solve the quadratic equation, we need to rearrange it into the standard form $ax^2 + bx + c = 0$ . We do this by moving all terms to one side of the equation and combining like terms.

Step 5: Apply the quadratic formula

Since the quadratic equation $2x^2 - 16x + 23 = 0$ cannot be easily factored, we use the quadratic formula to find the roots. Here, $a=2$ , $b=-16$ , and $c=23$ .

Step 6: Substitute values and simplify

Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula and simplify the expression. Remember to simplify the square root and then divide all terms by the common factor in the numerator and denominator.

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Find the roots of the equation 12x−3+1x−5=1\frac{1}{2x-3} + \frac{1}{x-5} = 12x−31​+x−51​=1 x≠32,5x \neq \frac{3}{2}, 5x=23​,5 (roots, equation, fractions, domain, quadratic)