Solve 5x^2-104=6x

$$5 x ^ { 2 } - 104 = 6 x$$

Solve for x

$$x=-4$$

$$x = \frac{26}{5} = 5\frac{1}{5} = 5.2$$

Steps Using Factoring By Grouping

Subtract 6x from both sides.

$$5x^{2}-104-6x=0$$

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

$$5x^{2}-6x-104=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx-104. To find a and b, set up a system to be solved.

$$a+b=-6 ab=5\left(-104\right)=-520$$

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 -520.

$$1,-520 2,-260 4,-130 5,-104 8,-65 10,-52 13,-40 20,-26$$

Calculate the sum for each pair.

$$1-520=-519 2-260=-258 4-130=-126 5-104=-99 8-65=-57 10-52=-42 13-40=-27 20-26=-6$$

The solution is the pair that gives sum -6.

$$a=-26 b=20$$

Rewrite 5x^{2}-6x-104 as \left(5x^{2}-26x\right)+\left(20x-104\right).

$$\left(5x^{2}-26x\right)+\left(20x-104\right)$$

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

$$x\left(5x-26\right)+4\left(5x-26\right)$$

Factor out common term 5x-26 by using distributive property.

$$\left(5x-26\right)\left(x+4\right)$$

To find equation solutions, solve 5x-26=0 and x+4=0.

$$x=\frac{26}{5} x=-4$$

Steps Using the Quadratic Formula

Subtract 6x from both sides.

$$5x^{2}-104-6x=0$$

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.

$$5x^{2}-6x-104=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -6 for b, and -104 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 5\left(-104\right)}}{2\times 5}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 5\left(-104\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-6\right)±\sqrt{36-20\left(-104\right)}}{2\times 5}$$

Multiply -20 times -104.

$$x=\frac{-\left(-6\right)±\sqrt{36+2080}}{2\times 5}$$

Add 36 to 2080.

$$x=\frac{-\left(-6\right)±\sqrt{2116}}{2\times 5}$$

Take the square root of 2116.

$$x=\frac{-\left(-6\right)±46}{2\times 5}$$

The opposite of -6 is 6.

$$x=\frac{6±46}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{6±46}{10}$$

Now solve the equation x=\frac{6±46}{10} when ± is plus. Add 6 to 46.

$$x=\frac{52}{10}$$

Reduce the fraction \frac{52}{10} to lowest terms by extracting and canceling out 2.

$$x=\frac{26}{5}$$

Now solve the equation x=\frac{6±46}{10} when ± is minus. Subtract 46 from 6.

$$x=-\frac{40}{10}$$

Divide -40 by 10.

$$x=-4$$

The equation is now solved.

$$x=\frac{26}{5} x=-4$$

Steps for Completing the Square

Subtract 6x from both sides.

$$5x^{2}-104-6x=0$$

Add 104 to both sides. Anything plus zero gives itself.

$$5x^{2}-6x=104$$

Divide both sides by 5.

$$\frac{5x^{2}-6x}{5}=\frac{104}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}-\frac{6}{5}x=\frac{104}{5}$$

Divide -\frac{6}{5}, the coefficient of the x term, by 2 to get -\frac{3}{5}. Then add the square of -\frac{3}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6}{5}x+\left(-\frac{3}{5}\right)^{2}=\frac{104}{5}+\left(-\frac{3}{5}\right)^{2}$$

Square -\frac{3}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6}{5}x+\frac{9}{25}=\frac{104}{5}+\frac{9}{25}$$

Add \frac{104}{5} to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{6}{5}x+\frac{9}{25}=\frac{529}{25}$$

Factor x^{2}-\frac{6}{5}x+\frac{9}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{5}\right)^{2}=\frac{529}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{5}\right)^{2}}=\sqrt{\frac{529}{25}}$$

Simplify.

$$x-\frac{3}{5}=\frac{23}{5} x-\frac{3}{5}=-\frac{23}{5}$$

Add \frac{3}{5} to both sides of the equation.

$$x=\frac{26}{5} x=-4$$