Solve 3x^2-4x+5(x^2-2)=3x(x-2)/2+14

$$3 x ^ { 2 } - 4 x + 5 ( x ^ { 2 } - 2 ) = \frac { 3 x ( x - 2 ) } { 2 } + 14$$

Solve for x

$$x = -\frac{24}{13} = -1\frac{11}{13} \approx -1.846153846$$

$$x=2$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 2.

$$6x^{2}-8x+10\left(x^{2}-2\right)=3x\left(x-2\right)+28$$

Use the distributive property to multiply 10 by x^{2}-2.

$$6x^{2}-8x+10x^{2}-20=3x\left(x-2\right)+28$$

Combine 6x^{2} and 10x^{2} to get 16x^{2}.

$$16x^{2}-8x-20=3x\left(x-2\right)+28$$

Use the distributive property to multiply 3x by x-2.

$$16x^{2}-8x-20=3x^{2}-6x+28$$

Subtract 3x^{2} from both sides.

$$16x^{2}-8x-20-3x^{2}=-6x+28$$

Combine 16x^{2} and -3x^{2} to get 13x^{2}.

$$13x^{2}-8x-20=-6x+28$$

Add 6x to both sides.

$$13x^{2}-8x-20+6x=28$$

Combine -8x and 6x to get -2x.

$$13x^{2}-2x-20=28$$

Subtract 28 from both sides.

$$13x^{2}-2x-20-28=0$$

Subtract 28 from -20 to get -48.

$$13x^{2}-2x-48=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 13x^{2}+ax+bx-48. To find a and b, set up a system to be solved.

$$a+b=-2 ab=13\left(-48\right)=-624$$

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 -624.

$$1,-624 2,-312 3,-208 4,-156 6,-104 8,-78 12,-52 13,-48 16,-39 24,-26$$

Calculate the sum for each pair.

$$1-624=-623 2-312=-310 3-208=-205 4-156=-152 6-104=-98 8-78=-70 12-52=-40 13-48=-35 16-39=-23 24-26=-2$$

The solution is the pair that gives sum -2.

$$a=-26 b=24$$

Rewrite 13x^{2}-2x-48 as \left(13x^{2}-26x\right)+\left(24x-48\right).

$$\left(13x^{2}-26x\right)+\left(24x-48\right)$$

Factor out 13x in the first and 24 in the second group.

$$13x\left(x-2\right)+24\left(x-2\right)$$

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

$$\left(x-2\right)\left(13x+24\right)$$

To find equation solutions, solve x-2=0 and 13x+24=0.

$$x=2 x=-\frac{24}{13}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$6x^{2}-8x+10\left(x^{2}-2\right)=3x\left(x-2\right)+28$$

Use the distributive property to multiply 10 by x^{2}-2.

$$6x^{2}-8x+10x^{2}-20=3x\left(x-2\right)+28$$

Combine 6x^{2} and 10x^{2} to get 16x^{2}.

$$16x^{2}-8x-20=3x\left(x-2\right)+28$$

Use the distributive property to multiply 3x by x-2.

$$16x^{2}-8x-20=3x^{2}-6x+28$$

Subtract 3x^{2} from both sides.

$$16x^{2}-8x-20-3x^{2}=-6x+28$$

Combine 16x^{2} and -3x^{2} to get 13x^{2}.

$$13x^{2}-8x-20=-6x+28$$

Add 6x to both sides.

$$13x^{2}-8x-20+6x=28$$

Combine -8x and 6x to get -2x.

$$13x^{2}-2x-20=28$$

Subtract 28 from both sides.

$$13x^{2}-2x-20-28=0$$

Subtract 28 from -20 to get -48.

$$13x^{2}-2x-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 13 for a, -2 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 13\left(-48\right)}}{2\times 13}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 13\left(-48\right)}}{2\times 13}$$

Multiply -4 times 13.

$$x=\frac{-\left(-2\right)±\sqrt{4-52\left(-48\right)}}{2\times 13}$$

Multiply -52 times -48.

$$x=\frac{-\left(-2\right)±\sqrt{4+2496}}{2\times 13}$$

Add 4 to 2496.

$$x=\frac{-\left(-2\right)±\sqrt{2500}}{2\times 13}$$

Take the square root of 2500.

$$x=\frac{-\left(-2\right)±50}{2\times 13}$$

The opposite of -2 is 2.

$$x=\frac{2±50}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{2±50}{26}$$

Now solve the equation x=\frac{2±50}{26} when ± is plus. Add 2 to 50.

$$x=\frac{52}{26}$$

Divide 52 by 26.

$$x=2$$

Now solve the equation x=\frac{2±50}{26} when ± is minus. Subtract 50 from 2.

$$x=-\frac{48}{26}$$

Reduce the fraction \frac{-48}{26} to lowest terms by extracting and canceling out 2.

$$x=-\frac{24}{13}$$

The equation is now solved.

$$x=2 x=-\frac{24}{13}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$6x^{2}-8x+10\left(x^{2}-2\right)=3x\left(x-2\right)+28$$

Use the distributive property to multiply 10 by x^{2}-2.

$$6x^{2}-8x+10x^{2}-20=3x\left(x-2\right)+28$$

Combine 6x^{2} and 10x^{2} to get 16x^{2}.

$$16x^{2}-8x-20=3x\left(x-2\right)+28$$

Use the distributive property to multiply 3x by x-2.

$$16x^{2}-8x-20=3x^{2}-6x+28$$

Subtract 3x^{2} from both sides.

$$16x^{2}-8x-20-3x^{2}=-6x+28$$

Combine 16x^{2} and -3x^{2} to get 13x^{2}.

$$13x^{2}-8x-20=-6x+28$$

Add 6x to both sides.

$$13x^{2}-8x-20+6x=28$$

Combine -8x and 6x to get -2x.

$$13x^{2}-2x-20=28$$

Add 20 to both sides.

$$13x^{2}-2x=28+20$$

Add 28 and 20 to get 48.

$$13x^{2}-2x=48$$

Divide both sides by 13.

$$\frac{13x^{2}-2x}{13}=\frac{48}{13}$$

Dividing by 13 undoes the multiplication by 13.

$$x^{2}-\frac{2}{13}x=\frac{48}{13}$$

Divide -\frac{2}{13}, the coefficient of the x term, by 2 to get -\frac{1}{13}. Then add the square of -\frac{1}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{13}x+\left(-\frac{1}{13}\right)^{2}=\frac{48}{13}+\left(-\frac{1}{13}\right)^{2}$$

Square -\frac{1}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{13}x+\frac{1}{169}=\frac{48}{13}+\frac{1}{169}$$

Add \frac{48}{13} to \frac{1}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{13}x+\frac{1}{169}=\frac{625}{169}$$

Factor x^{2}-\frac{2}{13}x+\frac{1}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{13}\right)^{2}=\frac{625}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{13}\right)^{2}}=\sqrt{\frac{625}{169}}$$

Simplify.

$$x-\frac{1}{13}=\frac{25}{13} x-\frac{1}{13}=-\frac{25}{13}$$

Add \frac{1}{13} to both sides of the equation.

$$x=2 x=-\frac{24}{13}$$