Solve 2x^2-x-5=43+3x

$$2 x ^ { 2 } - x - 5 = 43 + 3 x$$

Solve for x

$$x=-4$$

$$x=6$$

Steps Using Factoring By Grouping

Subtract 43 from both sides.

$$2x^{2}-x-5-43=3x$$

Subtract 43 from -5 to get -48.

$$2x^{2}-x-48=3x$$

Subtract 3x from both sides.

$$2x^{2}-x-48-3x=0$$

Combine -x and -3x to get -4x.

$$2x^{2}-4x-48=0$$

Divide both sides by 2.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-24\right)=-24$$

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 -24.

$$1,-24 2,-12 3,-8 4,-6$$

Calculate the sum for each pair.

$$1-24=-23 2-12=-10 3-8=-5 4-6=-2$$

The solution is the pair that gives sum -2.

$$a=-6 b=4$$

Rewrite x^{2}-2x-24 as \left(x^{2}-6x\right)+\left(4x-24\right).

$$\left(x^{2}-6x\right)+\left(4x-24\right)$$

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

$$x\left(x-6\right)+4\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(x+4\right)$$

To find equation solutions, solve x-6=0 and x+4=0.

$$x=6 x=-4$$

Steps Using the Quadratic Formula

Subtract 43 from both sides.

$$2x^{2}-x-5-43=3x$$

Subtract 43 from -5 to get -48.

$$2x^{2}-x-48=3x$$

Subtract 3x from both sides.

$$2x^{2}-x-48-3x=0$$

Combine -x and -3x to get -4x.

$$2x^{2}-4x-48=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -48 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-48\right)}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-48\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-48\right)}}{2\times 2}$$

Multiply -8 times -48.

$$x=\frac{-\left(-4\right)±\sqrt{16+384}}{2\times 2}$$

Add 16 to 384.

$$x=\frac{-\left(-4\right)±\sqrt{400}}{2\times 2}$$

Take the square root of 400.

$$x=\frac{-\left(-4\right)±20}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±20}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±20}{4}$$

Now solve the equation x=\frac{4±20}{4} when ± is plus. Add 4 to 20.

$$x=\frac{24}{4}$$

Divide 24 by 4.

$$x=6$$

Now solve the equation x=\frac{4±20}{4} when ± is minus. Subtract 20 from 4.

$$x=-\frac{16}{4}$$

Divide -16 by 4.

$$x=-4$$

The equation is now solved.

$$x=6 x=-4$$

Steps for Completing the Square

Subtract 3x from both sides.

$$2x^{2}-x-5-3x=43$$

Combine -x and -3x to get -4x.

$$2x^{2}-4x-5=43$$

Add 5 to both sides.

$$2x^{2}-4x=43+5$$

Add 43 and 5 to get 48.

$$2x^{2}-4x=48$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{48}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{48}{2}$$

Divide -4 by 2.

$$x^{2}-2x=\frac{48}{2}$$

Divide 48 by 2.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=24+1$$

Add 24 to 1.

$$x^{2}-2x+1=25$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=25$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{25}$$

Simplify.

$$x-1=5 x-1=-5$$

Add 1 to both sides of the equation.

$$x=6 x=-4$$