Solve 10x^2-6=-14x^2+10x+50

$$10 x ^ { 2 } - 6 = - 14 x ^ { 2 } + 10 x + 50$$

Solve for x

$$x = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

$$x = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

Steps Using Factoring By Grouping

Add 14x^{2} to both sides.

$$10x^{2}-6+14x^{2}=10x+50$$

Combine 10x^{2} and 14x^{2} to get 24x^{2}.

$$24x^{2}-6=10x+50$$

Subtract 10x from both sides.

$$24x^{2}-6-10x=50$$

Subtract 50 from both sides.

$$24x^{2}-6-10x-50=0$$

Subtract 50 from -6 to get -56.

$$24x^{2}-56-10x=0$$

Divide both sides by 2.

$$12x^{2}-28-5x=0$$

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

$$12x^{2}-5x-28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 12x^{2}+ax+bx-28. To find a and b, set up a system to be solved.

$$a+b=-5 ab=12\left(-28\right)=-336$$

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 -336.

$$1,-336 2,-168 3,-112 4,-84 6,-56 7,-48 8,-42 12,-28 14,-24 16,-21$$

Calculate the sum for each pair.

$$1-336=-335 2-168=-166 3-112=-109 4-84=-80 6-56=-50 7-48=-41 8-42=-34 12-28=-16 14-24=-10 16-21=-5$$

The solution is the pair that gives sum -5.

$$a=-21 b=16$$

Rewrite 12x^{2}-5x-28 as \left(12x^{2}-21x\right)+\left(16x-28\right).

$$\left(12x^{2}-21x\right)+\left(16x-28\right)$$

Factor out 3x in the first and 4 in the second group.

$$3x\left(4x-7\right)+4\left(4x-7\right)$$

Factor out common term 4x-7 by using distributive property.

$$\left(4x-7\right)\left(3x+4\right)$$

To find equation solutions, solve 4x-7=0 and 3x+4=0.

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

Steps Using the Quadratic Formula

Add 14x^{2} to both sides.

$$10x^{2}-6+14x^{2}=10x+50$$

Combine 10x^{2} and 14x^{2} to get 24x^{2}.

$$24x^{2}-6=10x+50$$

Subtract 10x from both sides.

$$24x^{2}-6-10x=50$$

Subtract 50 from both sides.

$$24x^{2}-6-10x-50=0$$

Subtract 50 from -6 to get -56.

$$24x^{2}-56-10x=0$$

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.

$$24x^{2}-10x-56=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, -10 for b, and -56 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 24\left(-56\right)}}{2\times 24}$$

Square -10.

$$x=\frac{-\left(-10\right)±\sqrt{100-4\times 24\left(-56\right)}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-\left(-10\right)±\sqrt{100-96\left(-56\right)}}{2\times 24}$$

Multiply -96 times -56.

$$x=\frac{-\left(-10\right)±\sqrt{100+5376}}{2\times 24}$$

Add 100 to 5376.

$$x=\frac{-\left(-10\right)±\sqrt{5476}}{2\times 24}$$

Take the square root of 5476.

$$x=\frac{-\left(-10\right)±74}{2\times 24}$$

The opposite of -10 is 10.

$$x=\frac{10±74}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{10±74}{48}$$

Now solve the equation x=\frac{10±74}{48} when ± is plus. Add 10 to 74.

$$x=\frac{84}{48}$$

Reduce the fraction \frac{84}{48} to lowest terms by extracting and canceling out 12.

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

Now solve the equation x=\frac{10±74}{48} when ± is minus. Subtract 74 from 10.

$$x=-\frac{64}{48}$$

Reduce the fraction \frac{-64}{48} to lowest terms by extracting and canceling out 16.

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

The equation is now solved.

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

Steps for Completing the Square

Add 14x^{2} to both sides.

$$10x^{2}-6+14x^{2}=10x+50$$

Combine 10x^{2} and 14x^{2} to get 24x^{2}.

$$24x^{2}-6=10x+50$$

Subtract 10x from both sides.

$$24x^{2}-6-10x=50$$

Add 6 to both sides.

$$24x^{2}-10x=50+6$$

Add 50 and 6 to get 56.

$$24x^{2}-10x=56$$

Divide both sides by 24.

$$\frac{24x^{2}-10x}{24}=\frac{56}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\left(-\frac{10}{24}\right)x=\frac{56}{24}$$

Reduce the fraction \frac{-10}{24} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{5}{12}x=\frac{56}{24}$$

Reduce the fraction \frac{56}{24} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{5}{12}x=\frac{7}{3}$$

Divide -\frac{5}{12}, the coefficient of the x term, by 2 to get -\frac{5}{24}. Then add the square of -\frac{5}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{12}x+\left(-\frac{5}{24}\right)^{2}=\frac{7}{3}+\left(-\frac{5}{24}\right)^{2}$$

Square -\frac{5}{24} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{12}x+\frac{25}{576}=\frac{7}{3}+\frac{25}{576}$$

Add \frac{7}{3} to \frac{25}{576} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{12}x+\frac{25}{576}=\frac{1369}{576}$$

Factor x^{2}-\frac{5}{12}x+\frac{25}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{24}\right)^{2}=\frac{1369}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{24}\right)^{2}}=\sqrt{\frac{1369}{576}}$$

Simplify.

$$x-\frac{5}{24}=\frac{37}{24} x-\frac{5}{24}=-\frac{37}{24}$$

Add \frac{5}{24} to both sides of the equation.

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