Solve 2x^2-6x/x-3=7

$$\frac { 2 x ^ { 2 } - 6 x } { x - 3 } = 7$$

Solve for x

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Steps Using Factoring By Grouping

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by x-3.

$$2x^{2}-6x=7\left(x-3\right)$$

Use the distributive property to multiply 7 by x-3.

$$2x^{2}-6x=7x-21$$

Subtract 7x from both sides.

$$2x^{2}-6x-7x=-21$$

Combine -6x and -7x to get -13x.

$$2x^{2}-13x=-21$$

Add 21 to both sides.

$$2x^{2}-13x+21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx+21. To find a and b, set up a system to be solved.

$$a+b=-13 ab=2\times 21=42$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 42.

$$-1,-42 -2,-21 -3,-14 -6,-7$$

Calculate the sum for each pair.

$$-1-42=-43 -2-21=-23 -3-14=-17 -6-7=-13$$

The solution is the pair that gives sum -13.

$$a=-7 b=-6$$

Rewrite 2x^{2}-13x+21 as \left(2x^{2}-7x\right)+\left(-6x+21\right).

$$\left(2x^{2}-7x\right)+\left(-6x+21\right)$$

Factor out x in the first and -3 in the second group.

$$x\left(2x-7\right)-3\left(2x-7\right)$$

Factor out common term 2x-7 by using distributive property.

$$\left(2x-7\right)\left(x-3\right)$$

To find equation solutions, solve 2x-7=0 and x-3=0.

$$x=\frac{7}{2} x=3$$

Variable x cannot be equal to 3.

$$x=\frac{7}{2}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by x-3.

$$2x^{2}-6x=7\left(x-3\right)$$

Use the distributive property to multiply 7 by x-3.

$$2x^{2}-6x=7x-21$$

Subtract 7x from both sides.

$$2x^{2}-6x-7x=-21$$

Combine -6x and -7x to get -13x.

$$2x^{2}-13x=-21$$

Add 21 to both sides.

$$2x^{2}-13x+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -13 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 2\times 21}}{2\times 2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 2\times 21}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-13\right)±\sqrt{169-8\times 21}}{2\times 2}$$

Multiply -8 times 21.

$$x=\frac{-\left(-13\right)±\sqrt{169-168}}{2\times 2}$$

Add 169 to -168.

$$x=\frac{-\left(-13\right)±\sqrt{1}}{2\times 2}$$

Take the square root of 1.

$$x=\frac{-\left(-13\right)±1}{2\times 2}$$

The opposite of -13 is 13.

$$x=\frac{13±1}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{13±1}{4}$$

Now solve the equation x=\frac{13±1}{4} when ± is plus. Add 13 to 1.

$$x=\frac{14}{4}$$

Reduce the fraction \frac{14}{4} to lowest terms by extracting and canceling out 2.

$$x=\frac{7}{2}$$

Now solve the equation x=\frac{13±1}{4} when ± is minus. Subtract 1 from 13.

$$x=\frac{12}{4}$$

Divide 12 by 4.

$$x=3$$

The equation is now solved.

$$x=\frac{7}{2} x=3$$

Variable x cannot be equal to 3.

$$x=\frac{7}{2}$$

Steps for Completing the Square

Variable x cannot be equal to 3 since division by zero is not defined. Multiply both sides of the equation by x-3.

$$2x^{2}-6x=7\left(x-3\right)$$

Use the distributive property to multiply 7 by x-3.

$$2x^{2}-6x=7x-21$$

Subtract 7x from both sides.

$$2x^{2}-6x-7x=-21$$

Combine -6x and -7x to get -13x.

$$2x^{2}-13x=-21$$

Divide both sides by 2.

$$\frac{2x^{2}-13x}{2}=-\frac{21}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{13}{2}x=-\frac{21}{2}$$

Divide -\frac{13}{2}, the coefficient of the x term, by 2 to get -\frac{13}{4}. Then add the square of -\frac{13}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{2}x+\left(-\frac{13}{4}\right)^{2}=-\frac{21}{2}+\left(-\frac{13}{4}\right)^{2}$$

Square -\frac{13}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=-\frac{21}{2}+\frac{169}{16}$$

Add -\frac{21}{2} to \frac{169}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{2}x+\frac{169}{16}=\frac{1}{16}$$

Factor x^{2}-\frac{13}{2}x+\frac{169}{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}.

$$\left(x-\frac{13}{4}\right)^{2}=\frac{1}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{4}\right)^{2}}=\sqrt{\frac{1}{16}}$$

Simplify.

$$x-\frac{13}{4}=\frac{1}{4} x-\frac{13}{4}=-\frac{1}{4}$$

Add \frac{13}{4} to both sides of the equation.

$$x=\frac{7}{2} x=3$$

Variable x cannot be equal to 3.

$$x=\frac{7}{2}$$