Solve left(3-xright)left(x^2+3x+9right)+sqrt{3}xleft(sqrt{3}-sqrt{12}xright)+left(x-2right)^2left(x+2right)=-13x+15

$$\left( 3-x \right) \left( { x }^{ 2 } +3x+9 \right) + \sqrt{ 3 } x \left( \sqrt{ 3 } - \sqrt{ 12 } x \right) + { \left(x-2 \right) }^{ 2 } \left( x+2 \right) = -13x+15$$

Solve for x

$$x=-1$$

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 3-x by x^{2}+3x+9 and combine like terms.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-\sqrt{12}x\right)+\left(x-2\right)^{2}\left(x+2\right)=-13x+15$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)+\left(x-2\right)^{2}\left(x+2\right)=-13x+15$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)+\left(x^{2}-4x+4\right)\left(x+2\right)=-13x+15$$

Use the distributive property to multiply x^{2}-4x+4 by x+2 and combine like terms.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)+x^{3}-2x^{2}-4x+8=-13x+15$$

Combine -x^{3} and x^{3} to get 0.

$$27+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}-4x+8=-13x+15$$

Add 27 and 8 to get 35.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}-4x=-13x+15$$

Add 13x to both sides.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}-4x+13x=15$$

Combine -4x and 13x to get 9x.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}+9x=15$$

Subtract 15 from both sides.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}+9x-15=0$$

Subtract 15 from 35 to get 20.

$$20+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}+9x=0$$

Use the distributive property to multiply \sqrt{3}x by \sqrt{3}-2\sqrt{3}x.

$$20+x\left(\sqrt{3}\right)^{2}-2\left(\sqrt{3}\right)^{2}x^{2}-2x^{2}+9x=0$$

The square of \sqrt{3} is 3.

$$20+x\times 3-2\left(\sqrt{3}\right)^{2}x^{2}-2x^{2}+9x=0$$

The square of \sqrt{3} is 3.

$$20+x\times 3-2\times 3x^{2}-2x^{2}+9x=0$$

Multiply -2 and 3 to get -6.

$$20+x\times 3-6x^{2}-2x^{2}+9x=0$$

Combine -6x^{2} and -2x^{2} to get -8x^{2}.

$$20+x\times 3-8x^{2}+9x=0$$

Combine x\times 3 and 9x to get 12x.

$$20+12x-8x^{2}=0$$

Divide both sides by 4.

$$5+3x-2x^{2}=0$$

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

$$-2x^{2}+3x+5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=3 ab=-2\times 5=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -10.

$$-1,10 -2,5$$

Calculate the sum for each pair.

$$-1+10=9 -2+5=3$$

The solution is the pair that gives sum 3.

$$a=5 b=-2$$

Rewrite -2x^{2}+3x+5 as \left(-2x^{2}+5x\right)+\left(-2x+5\right).

$$\left(-2x^{2}+5x\right)+\left(-2x+5\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(2x-5\right)-\left(2x-5\right)$$

Factor out common term 2x-5 by using distributive property.

$$\left(2x-5\right)\left(-x-1\right)$$

To find equation solutions, solve 2x-5=0 and -x-1=0.

$$x=\frac{5}{2} x=-1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3-x by x^{2}+3x+9 and combine like terms.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-\sqrt{12}x\right)+\left(x-2\right)^{2}\left(x+2\right)=-13x+15$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)+\left(x-2\right)^{2}\left(x+2\right)=-13x+15$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)+\left(x^{2}-4x+4\right)\left(x+2\right)=-13x+15$$

Use the distributive property to multiply x^{2}-4x+4 by x+2 and combine like terms.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)+x^{3}-2x^{2}-4x+8=-13x+15$$

Combine -x^{3} and x^{3} to get 0.

$$27+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}-4x+8=-13x+15$$

Add 27 and 8 to get 35.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}-4x=-13x+15$$

Add 13x to both sides.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}-4x+13x=15$$

Combine -4x and 13x to get 9x.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}+9x=15$$

Subtract 15 from both sides.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}+9x-15=0$$

Subtract 15 from 35 to get 20.

$$20+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}+9x=0$$

Use the distributive property to multiply \sqrt{3}x by \sqrt{3}-2\sqrt{3}x.

$$20+x\left(\sqrt{3}\right)^{2}-2\left(\sqrt{3}\right)^{2}x^{2}-2x^{2}+9x=0$$

The square of \sqrt{3} is 3.

$$20+x\times 3-2\left(\sqrt{3}\right)^{2}x^{2}-2x^{2}+9x=0$$

The square of \sqrt{3} is 3.

$$20+x\times 3-2\times 3x^{2}-2x^{2}+9x=0$$

Multiply -2 and 3 to get -6.

$$20+x\times 3-6x^{2}-2x^{2}+9x=0$$

Combine -6x^{2} and -2x^{2} to get -8x^{2}.

$$20+x\times 3-8x^{2}+9x=0$$

Combine x\times 3 and 9x to get 12x.

$$20+12x-8x^{2}=0$$

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.

$$-8x^{2}+12x+20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -8 for a, 12 for b, and 20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-8\right)\times 20}}{2\left(-8\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-8\right)\times 20}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$x=\frac{-12±\sqrt{144+32\times 20}}{2\left(-8\right)}$$

Multiply 32 times 20.

$$x=\frac{-12±\sqrt{144+640}}{2\left(-8\right)}$$

Add 144 to 640.

$$x=\frac{-12±\sqrt{784}}{2\left(-8\right)}$$

Take the square root of 784.

$$x=\frac{-12±28}{2\left(-8\right)}$$

Multiply 2 times -8.

$$x=\frac{-12±28}{-16}$$

Now solve the equation x=\frac{-12±28}{-16} when ± is plus. Add -12 to 28.

$$x=\frac{16}{-16}$$

Divide 16 by -16.

$$x=-1$$

Now solve the equation x=\frac{-12±28}{-16} when ± is minus. Subtract 28 from -12.

$$x=-\frac{40}{-16}$$

Reduce the fraction \frac{-40}{-16} to lowest terms by extracting and canceling out 8.

$$x=\frac{5}{2}$$

The equation is now solved.

$$x=-1 x=\frac{5}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 3-x by x^{2}+3x+9 and combine like terms.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-\sqrt{12}x\right)+\left(x-2\right)^{2}\left(x+2\right)=-13x+15$$

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)+\left(x-2\right)^{2}\left(x+2\right)=-13x+15$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-2\right)^{2}.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)+\left(x^{2}-4x+4\right)\left(x+2\right)=-13x+15$$

Use the distributive property to multiply x^{2}-4x+4 by x+2 and combine like terms.

$$27-x^{3}+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)+x^{3}-2x^{2}-4x+8=-13x+15$$

Combine -x^{3} and x^{3} to get 0.

$$27+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}-4x+8=-13x+15$$

Add 27 and 8 to get 35.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}-4x=-13x+15$$

Add 13x to both sides.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}-4x+13x=15$$

Combine -4x and 13x to get 9x.

$$35+\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}+9x=15$$

Subtract 35 from both sides.

$$\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}+9x=15-35$$

Subtract 35 from 15 to get -20.

$$\sqrt{3}x\left(\sqrt{3}-2\sqrt{3}x\right)-2x^{2}+9x=-20$$

Use the distributive property to multiply \sqrt{3}x by \sqrt{3}-2\sqrt{3}x.

$$x\left(\sqrt{3}\right)^{2}-2\left(\sqrt{3}\right)^{2}x^{2}-2x^{2}+9x=-20$$

The square of \sqrt{3} is 3.

$$x\times 3-2\left(\sqrt{3}\right)^{2}x^{2}-2x^{2}+9x=-20$$

The square of \sqrt{3} is 3.

$$x\times 3-2\times 3x^{2}-2x^{2}+9x=-20$$

Multiply -2 and 3 to get -6.

$$x\times 3-6x^{2}-2x^{2}+9x=-20$$

Combine -6x^{2} and -2x^{2} to get -8x^{2}.

$$x\times 3-8x^{2}+9x=-20$$

Combine x\times 3 and 9x to get 12x.

$$12x-8x^{2}=-20$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-8x^{2}+12x=-20$$

Divide both sides by -8.

$$\frac{-8x^{2}+12x}{-8}=-\frac{20}{-8}$$

Dividing by -8 undoes the multiplication by -8.

$$x^{2}+\frac{12}{-8}x=-\frac{20}{-8}$$

Reduce the fraction \frac{12}{-8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{3}{2}x=-\frac{20}{-8}$$

Reduce the fraction \frac{-20}{-8} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{3}{2}x=\frac{5}{2}$$

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.

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

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

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{5}{2}+\frac{9}{16}$$

Add \frac{5}{2} to \frac{9}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{49}{16}$$

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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

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

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

$$x=\frac{5}{2} x=-1$$