Solve 2x(2x+1)-10x-16=5x^2

$$2 x ( 2 x + 1 ) - 10 x - 16 = 5 x ^ { 2 }$$

Solve for x

$$x=-4$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 2x by 2x+1.

$$4x^{2}+2x-10x-16=5x^{2}$$

Combine 2x and -10x to get -8x.

$$4x^{2}-8x-16=5x^{2}$$

Subtract 5x^{2} from both sides.

$$4x^{2}-8x-16-5x^{2}=0$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}-8x-16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-\left(-16\right)=16$$

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 16.

$$-1,-16 -2,-8 -4,-4$$

Calculate the sum for each pair.

$$-1-16=-17 -2-8=-10 -4-4=-8$$

The solution is the pair that gives sum -8.

$$a=-4 b=-4$$

Rewrite -x^{2}-8x-16 as \left(-x^{2}-4x\right)+\left(-4x-16\right).

$$\left(-x^{2}-4x\right)+\left(-4x-16\right)$$

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

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

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

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

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

$$x=-4 x=-4$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x by 2x+1.

$$4x^{2}+2x-10x-16=5x^{2}$$

Combine 2x and -10x to get -8x.

$$4x^{2}-8x-16=5x^{2}$$

Subtract 5x^{2} from both sides.

$$4x^{2}-8x-16-5x^{2}=0$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}-8x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -8 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4\left(-16\right)}}{2\left(-1\right)}$$

Multiply 4 times -16.

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

Add 64 to -64.

$$x=\frac{-\left(-8\right)±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

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

The opposite of -8 is 8.

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

Multiply 2 times -1.

$$x=\frac{8}{-2}$$

Divide 8 by -2.

$$x=-4$$

Steps for Completing the Square

Use the distributive property to multiply 2x by 2x+1.

$$4x^{2}+2x-10x-16=5x^{2}$$

Combine 2x and -10x to get -8x.

$$4x^{2}-8x-16=5x^{2}$$

Subtract 5x^{2} from both sides.

$$4x^{2}-8x-16-5x^{2}=0$$

Combine 4x^{2} and -5x^{2} to get -x^{2}.

$$-x^{2}-8x-16=0$$

Add 16 to both sides. Anything plus zero gives itself.

$$-x^{2}-8x=16$$

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{8}{-1}\right)x=\frac{16}{-1}$$

Divide -8 by -1.

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

Divide 16 by -1.

$$x^{2}+8x=-16$$

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

$$x^{2}+8x+4^{2}=-16+4^{2}$$

Square 4.

$$x^{2}+8x+16=-16+16$$

Add -16 to 16.

$$x^{2}+8x+16=0$$

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

$$\left(x+4\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{0}$$

Simplify.

$$x+4=0 x+4=0$$

Subtract 4 from both sides of the equation.

$$x=-4 x=-4$$

The equation is now solved. Solutions are the same.

$$x=-4$$