Solve 5x-1/4x+3=7x-2/x-3

$$\frac { 5 x - 1 } { 4 x + 3 } = \frac { 7 x - 2 } { x - 3 }$$

Solve for x

$$x=\frac{\sqrt{1669}-29}{46}\approx 0.257682531$$

$$x=\frac{-\sqrt{1669}-29}{46}\approx -1.518552096$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -\frac{3}{4},3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(4x+3\right), the least common multiple of 4x+3,x-3.

$$\left(x-3\right)\left(5x-1\right)=\left(4x+3\right)\left(7x-2\right)$$

Use the distributive property to multiply x-3 by 5x-1 and combine like terms.

$$5x^{2}-16x+3=\left(4x+3\right)\left(7x-2\right)$$

Use the distributive property to multiply 4x+3 by 7x-2 and combine like terms.

$$5x^{2}-16x+3=28x^{2}+13x-6$$

Subtract 28x^{2} from both sides.

$$5x^{2}-16x+3-28x^{2}=13x-6$$

Combine 5x^{2} and -28x^{2} to get -23x^{2}.

$$-23x^{2}-16x+3=13x-6$$

Subtract 13x from both sides.

$$-23x^{2}-16x+3-13x=-6$$

Combine -16x and -13x to get -29x.

$$-23x^{2}-29x+3=-6$$

Add 6 to both sides.

$$-23x^{2}-29x+3+6=0$$

Add 3 and 6 to get 9.

$$-23x^{2}-29x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -23 for a, -29 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\left(-23\right)\times 9}}{2\left(-23\right)}$$

Square -29.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\left(-23\right)\times 9}}{2\left(-23\right)}$$

Multiply -4 times -23.

$$x=\frac{-\left(-29\right)±\sqrt{841+92\times 9}}{2\left(-23\right)}$$

Multiply 92 times 9.

$$x=\frac{-\left(-29\right)±\sqrt{841+828}}{2\left(-23\right)}$$

Add 841 to 828.

$$x=\frac{-\left(-29\right)±\sqrt{1669}}{2\left(-23\right)}$$

The opposite of -29 is 29.

$$x=\frac{29±\sqrt{1669}}{2\left(-23\right)}$$

Multiply 2 times -23.

$$x=\frac{29±\sqrt{1669}}{-46}$$

Now solve the equation x=\frac{29±\sqrt{1669}}{-46} when ± is plus. Add 29 to \sqrt{1669}.

$$x=\frac{\sqrt{1669}+29}{-46}$$

Divide 29+\sqrt{1669} by -46.

$$x=\frac{-\sqrt{1669}-29}{46}$$

Now solve the equation x=\frac{29±\sqrt{1669}}{-46} when ± is minus. Subtract \sqrt{1669} from 29.

$$x=\frac{29-\sqrt{1669}}{-46}$$

Divide 29-\sqrt{1669} by -46.

$$x=\frac{\sqrt{1669}-29}{46}$$

The equation is now solved.

$$x=\frac{-\sqrt{1669}-29}{46} x=\frac{\sqrt{1669}-29}{46}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -\frac{3}{4},3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(4x+3\right), the least common multiple of 4x+3,x-3.

$$\left(x-3\right)\left(5x-1\right)=\left(4x+3\right)\left(7x-2\right)$$

Use the distributive property to multiply x-3 by 5x-1 and combine like terms.

$$5x^{2}-16x+3=\left(4x+3\right)\left(7x-2\right)$$

Use the distributive property to multiply 4x+3 by 7x-2 and combine like terms.

$$5x^{2}-16x+3=28x^{2}+13x-6$$

Subtract 28x^{2} from both sides.

$$5x^{2}-16x+3-28x^{2}=13x-6$$

Combine 5x^{2} and -28x^{2} to get -23x^{2}.

$$-23x^{2}-16x+3=13x-6$$

Subtract 13x from both sides.

$$-23x^{2}-16x+3-13x=-6$$

Combine -16x and -13x to get -29x.

$$-23x^{2}-29x+3=-6$$

Subtract 3 from both sides.

$$-23x^{2}-29x=-6-3$$

Subtract 3 from -6 to get -9.

$$-23x^{2}-29x=-9$$

Divide both sides by -23.

$$\frac{-23x^{2}-29x}{-23}=-\frac{9}{-23}$$

Dividing by -23 undoes the multiplication by -23.

$$x^{2}+\left(-\frac{29}{-23}\right)x=-\frac{9}{-23}$$

Divide -29 by -23.

$$x^{2}+\frac{29}{23}x=-\frac{9}{-23}$$

Divide -9 by -23.

$$x^{2}+\frac{29}{23}x=\frac{9}{23}$$

Divide \frac{29}{23}, the coefficient of the x term, by 2 to get \frac{29}{46}. Then add the square of \frac{29}{46} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{29}{23}x+\left(\frac{29}{46}\right)^{2}=\frac{9}{23}+\left(\frac{29}{46}\right)^{2}$$

Square \frac{29}{46} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{29}{23}x+\frac{841}{2116}=\frac{9}{23}+\frac{841}{2116}$$

Add \frac{9}{23} to \frac{841}{2116} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{29}{23}x+\frac{841}{2116}=\frac{1669}{2116}$$

Factor x^{2}+\frac{29}{23}x+\frac{841}{2116}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{29}{46}\right)^{2}=\frac{1669}{2116}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{29}{46}\right)^{2}}=\sqrt{\frac{1669}{2116}}$$

Simplify.

$$x+\frac{29}{46}=\frac{\sqrt{1669}}{46} x+\frac{29}{46}=-\frac{\sqrt{1669}}{46}$$

Subtract \frac{29}{46} from both sides of the equation.

$$x=\frac{\sqrt{1669}-29}{46} x=\frac{-\sqrt{1669}-29}{46}$$