Solve -13x^2-9x+4=5+2x^2-x

$$- 13 x ^ { 2 } - 9 x + 4 = 5 + 2 x ^ { 2 } - x$$

Solve for x

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

$$x=-\frac{1}{5}=-0.2$$

Steps Using Factoring By Grouping

Subtract 5 from both sides.

$$-13x^{2}-9x+4-5=2x^{2}-x$$

Subtract 5 from 4 to get -1.

$$-13x^{2}-9x-1=2x^{2}-x$$

Subtract 2x^{2} from both sides.

$$-13x^{2}-9x-1-2x^{2}=-x$$

Combine -13x^{2} and -2x^{2} to get -15x^{2}.

$$-15x^{2}-9x-1=-x$$

Add x to both sides.

$$-15x^{2}-9x-1+x=0$$

Combine -9x and x to get -8x.

$$-15x^{2}-8x-1=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -15x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-15\left(-1\right)=15$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 15.

$$-1,-15 -3,-5$$

Calculate the sum for each pair.

$$-1-15=-16 -3-5=-8$$

The solution is the pair that gives sum -8.

$$a=-3 b=-5$$

Rewrite -15x^{2}-8x-1 as \left(-15x^{2}-3x\right)+\left(-5x-1\right).

$$\left(-15x^{2}-3x\right)+\left(-5x-1\right)$$

Factor out 3x in -15x^{2}-3x.

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

Factor out common term -5x-1 by using distributive property.

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

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

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

Steps Using the Quadratic Formula

Subtract 5 from both sides.

$$-13x^{2}-9x+4-5=2x^{2}-x$$

Subtract 5 from 4 to get -1.

$$-13x^{2}-9x-1=2x^{2}-x$$

Subtract 2x^{2} from both sides.

$$-13x^{2}-9x-1-2x^{2}=-x$$

Combine -13x^{2} and -2x^{2} to get -15x^{2}.

$$-15x^{2}-9x-1=-x$$

Add x to both sides.

$$-15x^{2}-9x-1+x=0$$

Combine -9x and x to get -8x.

$$-15x^{2}-8x-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, -8 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-15\right)\left(-1\right)}}{2\left(-15\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-15\right)\left(-1\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-\left(-8\right)±\sqrt{64+60\left(-1\right)}}{2\left(-15\right)}$$

Multiply 60 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64-60}}{2\left(-15\right)}$$

Add 64 to -60.

$$x=\frac{-\left(-8\right)±\sqrt{4}}{2\left(-15\right)}$$

Take the square root of 4.

$$x=\frac{-\left(-8\right)±2}{2\left(-15\right)}$$

The opposite of -8 is 8.

$$x=\frac{8±2}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{8±2}{-30}$$

Now solve the equation x=\frac{8±2}{-30} when ± is plus. Add 8 to 2.

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

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{8±2}{-30} when ± is minus. Subtract 2 from 8.

$$x=\frac{6}{-30}$$

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

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

The equation is now solved.

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

Steps for Completing the Square

Subtract 2x^{2} from both sides.

$$-13x^{2}-9x+4-2x^{2}=5-x$$

Combine -13x^{2} and -2x^{2} to get -15x^{2}.

$$-15x^{2}-9x+4=5-x$$

Add x to both sides.

$$-15x^{2}-9x+4+x=5$$

Combine -9x and x to get -8x.

$$-15x^{2}-8x+4=5$$

Subtract 4 from both sides.

$$-15x^{2}-8x=5-4$$

Subtract 4 from 5 to get 1.

$$-15x^{2}-8x=1$$

Divide both sides by -15.

$$\frac{-15x^{2}-8x}{-15}=\frac{1}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\left(-\frac{8}{-15}\right)x=\frac{1}{-15}$$

Divide -8 by -15.

$$x^{2}+\frac{8}{15}x=\frac{1}{-15}$$

Divide 1 by -15.

$$x^{2}+\frac{8}{15}x=-\frac{1}{15}$$

Divide \frac{8}{15}, the coefficient of the x term, by 2 to get \frac{4}{15}. Then add the square of \frac{4}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{8}{15}x+\left(\frac{4}{15}\right)^{2}=-\frac{1}{15}+\left(\frac{4}{15}\right)^{2}$$

Square \frac{4}{15} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{8}{15}x+\frac{16}{225}=-\frac{1}{15}+\frac{16}{225}$$

Add -\frac{1}{15} to \frac{16}{225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{8}{15}x+\frac{16}{225}=\frac{1}{225}$$

Factor x^{2}+\frac{8}{15}x+\frac{16}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{4}{15}\right)^{2}=\frac{1}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4}{15}\right)^{2}}=\sqrt{\frac{1}{225}}$$

Simplify.

$$x+\frac{4}{15}=\frac{1}{15} x+\frac{4}{15}=-\frac{1}{15}$$

Subtract \frac{4}{15} from both sides of the equation.

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