Solve x^2+6/3-7/2=x+15/2

$$\frac{ { x }^{ 2 } +6 }{ 3 } - \frac{ 7 }{ 2 } = \frac{ x+15 }{ 2 }$$

Solve for x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$x=6$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2\left(x^{2}+6\right)-21=3\left(x+15\right)$$

Use the distributive property to multiply 2 by x^{2}+6.

$$2x^{2}+12-21=3\left(x+15\right)$$

Subtract 21 from 12 to get -9.

$$2x^{2}-9=3\left(x+15\right)$$

Use the distributive property to multiply 3 by x+15.

$$2x^{2}-9=3x+45$$

Subtract 3x from both sides.

$$2x^{2}-9-3x=45$$

Subtract 45 from both sides.

$$2x^{2}-9-3x-45=0$$

Subtract 45 from -9 to get -54.

$$2x^{2}-54-3x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$2x^{2}-3x-54=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-54. To find a and b, set up a system to be solved.

$$a+b=-3 ab=2\left(-54\right)=-108$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -108.

$$1,-108 2,-54 3,-36 4,-27 6,-18 9,-12$$

Calculate the sum for each pair.

$$1-108=-107 2-54=-52 3-36=-33 4-27=-23 6-18=-12 9-12=-3$$

The solution is the pair that gives sum -3.

$$a=-12 b=9$$

Rewrite 2x^{2}-3x-54 as \left(2x^{2}-12x\right)+\left(9x-54\right).

$$\left(2x^{2}-12x\right)+\left(9x-54\right)$$

Factor out 2x in the first and 9 in the second group.

$$2x\left(x-6\right)+9\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(2x+9\right)$$

To find equation solutions, solve x-6=0 and 2x+9=0.

$$x=6 x=-\frac{9}{2}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2\left(x^{2}+6\right)-21=3\left(x+15\right)$$

Use the distributive property to multiply 2 by x^{2}+6.

$$2x^{2}+12-21=3\left(x+15\right)$$

Subtract 21 from 12 to get -9.

$$2x^{2}-9=3\left(x+15\right)$$

Use the distributive property to multiply 3 by x+15.

$$2x^{2}-9=3x+45$$

Subtract 3x from both sides.

$$2x^{2}-9-3x=45$$

Subtract 45 from both sides.

$$2x^{2}-9-3x-45=0$$

Subtract 45 from -9 to get -54.

$$2x^{2}-54-3x=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$2x^{2}-3x-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -3 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-54\right)}}{2\times 2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-54\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-54\right)}}{2\times 2}$$

Multiply -8 times -54.

$$x=\frac{-\left(-3\right)±\sqrt{9+432}}{2\times 2}$$

Add 9 to 432.

$$x=\frac{-\left(-3\right)±\sqrt{441}}{2\times 2}$$

Take the square root of 441.

$$x=\frac{-\left(-3\right)±21}{2\times 2}$$

The opposite of -3 is 3.

$$x=\frac{3±21}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{3±21}{4}$$

Now solve the equation x=\frac{3±21}{4} when ± is plus. Add 3 to 21.

$$x=\frac{24}{4}$$

Divide 24 by 4.

$$x=6$$

Now solve the equation x=\frac{3±21}{4} when ± is minus. Subtract 21 from 3.

$$x=-\frac{18}{4}$$

Reduce the fraction \frac{-18}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{9}{2}$$

The equation is now solved.

$$x=6 x=-\frac{9}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2\left(x^{2}+6\right)-21=3\left(x+15\right)$$

Use the distributive property to multiply 2 by x^{2}+6.

$$2x^{2}+12-21=3\left(x+15\right)$$

Subtract 21 from 12 to get -9.

$$2x^{2}-9=3\left(x+15\right)$$

Use the distributive property to multiply 3 by x+15.

$$2x^{2}-9=3x+45$$

Subtract 3x from both sides.

$$2x^{2}-9-3x=45$$

Add 9 to both sides.

$$2x^{2}-3x=45+9$$

Add 45 and 9 to get 54.

$$2x^{2}-3x=54$$

Divide both sides by 2.

$$\frac{2x^{2}-3x}{2}=\frac{54}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{3}{2}x=\frac{54}{2}$$

Divide 54 by 2.

$$x^{2}-\frac{3}{2}x=27$$

Divide -\frac{3}{2}, the coefficient of the x term, by 2 to get -\frac{3}{4}. Then add the square of -\frac{3}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=27+\left(-\frac{3}{4}\right)^{2}$$

Square -\frac{3}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=27+\frac{9}{16}$$

Add 27 to \frac{9}{16}.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{441}{16}$$

Factor x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{441}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Simplify.

$$x-\frac{3}{4}=\frac{21}{4} x-\frac{3}{4}=-\frac{21}{4}$$

Add \frac{3}{4} to both sides of the equation.

$$x=6 x=-\frac{9}{2}$$