Solve 15x^2-6=-13x

$$15 x ^ { 2 } - 6 = - 13 x$$

Solve for x

$$x = -\frac{6}{5} = -1\frac{1}{5} = -1.2$$

$$x=\frac{1}{3}\approx 0.333333333$$

Steps Using Factoring By Grouping

Add 13x to both sides.

$$15x^{2}-6+13x=0$$

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

$$15x^{2}+13x-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 15x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=13 ab=15\left(-6\right)=-90$$

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 -90.

$$-1,90 -2,45 -3,30 -5,18 -6,15 -9,10$$

Calculate the sum for each pair.

$$-1+90=89 -2+45=43 -3+30=27 -5+18=13 -6+15=9 -9+10=1$$

The solution is the pair that gives sum 13.

$$a=-5 b=18$$

Rewrite 15x^{2}+13x-6 as \left(15x^{2}-5x\right)+\left(18x-6\right).

$$\left(15x^{2}-5x\right)+\left(18x-6\right)$$

Factor out 5x in the first and 6 in the second group.

$$5x\left(3x-1\right)+6\left(3x-1\right)$$

Factor out common term 3x-1 by using distributive property.

$$\left(3x-1\right)\left(5x+6\right)$$

To find equation solutions, solve 3x-1=0 and 5x+6=0.

$$x=\frac{1}{3} x=-\frac{6}{5}$$

Steps Using the Quadratic Formula

Add 13x to both sides.

$$15x^{2}-6+13x=0$$

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.

$$15x^{2}+13x-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, 13 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 15\left(-6\right)}}{2\times 15}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 15\left(-6\right)}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-13±\sqrt{169-60\left(-6\right)}}{2\times 15}$$

Multiply -60 times -6.

$$x=\frac{-13±\sqrt{169+360}}{2\times 15}$$

Add 169 to 360.

$$x=\frac{-13±\sqrt{529}}{2\times 15}$$

Take the square root of 529.

$$x=\frac{-13±23}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{-13±23}{30}$$

Now solve the equation x=\frac{-13±23}{30} when ± is plus. Add -13 to 23.

$$x=\frac{10}{30}$$

Reduce the fraction \frac{10}{30} to lowest terms by extracting and canceling out 10.

$$x=\frac{1}{3}$$

Now solve the equation x=\frac{-13±23}{30} when ± is minus. Subtract 23 from -13.

$$x=-\frac{36}{30}$$

Reduce the fraction \frac{-36}{30} to lowest terms by extracting and canceling out 6.

$$x=-\frac{6}{5}$$

The equation is now solved.

$$x=\frac{1}{3} x=-\frac{6}{5}$$

Steps for Completing the Square

Add 13x to both sides.

$$15x^{2}-6+13x=0$$

Add 6 to both sides. Anything plus zero gives itself.

$$15x^{2}+13x=6$$

Divide both sides by 15.

$$\frac{15x^{2}+13x}{15}=\frac{6}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\frac{13}{15}x=\frac{6}{15}$$

Reduce the fraction \frac{6}{15} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{13}{15}x=\frac{2}{5}$$

Divide \frac{13}{15}, the coefficient of the x term, by 2 to get \frac{13}{30}. Then add the square of \frac{13}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{13}{15}x+\left(\frac{13}{30}\right)^{2}=\frac{2}{5}+\left(\frac{13}{30}\right)^{2}$$

Square \frac{13}{30} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{13}{15}x+\frac{169}{900}=\frac{2}{5}+\frac{169}{900}$$

Add \frac{2}{5} to \frac{169}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{13}{15}x+\frac{169}{900}=\frac{529}{900}$$

Factor x^{2}+\frac{13}{15}x+\frac{169}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{13}{30}\right)^{2}=\frac{529}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{30}\right)^{2}}=\sqrt{\frac{529}{900}}$$

Simplify.

$$x+\frac{13}{30}=\frac{23}{30} x+\frac{13}{30}=-\frac{23}{30}$$

Subtract \frac{13}{30} from both sides of the equation.

$$x=\frac{1}{3} x=-\frac{6}{5}$$