Solve 5x-7/x-3=4x-3/x

$$\frac{ 5x-7 }{ x-3 } = \frac{ 4x-3 }{ x }$$

Solve for x

$$x=-9$$

$$x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x.

$$x\left(5x-7\right)=\left(x-3\right)\left(4x-3\right)$$

Use the distributive property to multiply x by 5x-7.

$$5x^{2}-7x=\left(x-3\right)\left(4x-3\right)$$

Use the distributive property to multiply x-3 by 4x-3 and combine like terms.

$$5x^{2}-7x=4x^{2}-15x+9$$

Subtract 4x^{2} from both sides.

$$5x^{2}-7x-4x^{2}=-15x+9$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$x^{2}-7x=-15x+9$$

Add 15x to both sides.

$$x^{2}-7x+15x=9$$

Combine -7x and 15x to get 8x.

$$x^{2}+8x=9$$

Subtract 9 from both sides.

$$x^{2}+8x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-9\right)}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-9\right)}}{2}$$

Multiply -4 times -9.

$$x=\frac{-8±\sqrt{64+36}}{2}$$

Add 64 to 36.

$$x=\frac{-8±\sqrt{100}}{2}$$

Take the square root of 100.

$$x=\frac{-8±10}{2}$$

Now solve the equation x=\frac{-8±10}{2} when ± is plus. Add -8 to 10.

$$x=\frac{2}{2}$$

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-8±10}{2} when ± is minus. Subtract 10 from -8.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=1 x=-9$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,3 since division by zero is not defined. Multiply both sides of the equation by x\left(x-3\right), the least common multiple of x-3,x.

$$x\left(5x-7\right)=\left(x-3\right)\left(4x-3\right)$$

Use the distributive property to multiply x by 5x-7.

$$5x^{2}-7x=\left(x-3\right)\left(4x-3\right)$$

Use the distributive property to multiply x-3 by 4x-3 and combine like terms.

$$5x^{2}-7x=4x^{2}-15x+9$$

Subtract 4x^{2} from both sides.

$$5x^{2}-7x-4x^{2}=-15x+9$$

Combine 5x^{2} and -4x^{2} to get x^{2}.

$$x^{2}-7x=-15x+9$$

Add 15x to both sides.

$$x^{2}-7x+15x=9$$

Combine -7x and 15x to get 8x.

$$x^{2}+8x=9$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=9+4^{2}$$

Square 4.

$$x^{2}+8x+16=9+16$$

Add 9 to 16.

$$x^{2}+8x+16=25$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{25}$$

Simplify.

$$x+4=5 x+4=-5$$

Subtract 4 from both sides of the equation.

$$x=1 x=-9$$