Solve x^2+2x+28=3x^2+x

$$x ^ { 2 } + 2 x + 28 = 3 x ^ { 2 } + x$$

Solve for x

$$x = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$x=4$$

Steps Using Factoring By Grouping

Subtract 3x^{2} from both sides.

$$x^{2}+2x+28-3x^{2}=x$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}+2x+28=x$$

Subtract x from both sides.

$$-2x^{2}+2x+28-x=0$$

Combine 2x and -x to get x.

$$-2x^{2}+x+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -2x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=1 ab=-2\times 28=-56$$

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 -56.

$$-1,56 -2,28 -4,14 -7,8$$

Calculate the sum for each pair.

$$-1+56=55 -2+28=26 -4+14=10 -7+8=1$$

The solution is the pair that gives sum 1.

$$a=8 b=-7$$

Rewrite -2x^{2}+x+28 as \left(-2x^{2}+8x\right)+\left(-7x+28\right).

$$\left(-2x^{2}+8x\right)+\left(-7x+28\right)$$

Factor out 2x in the first and 7 in the second group.

$$2x\left(-x+4\right)+7\left(-x+4\right)$$

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

$$\left(-x+4\right)\left(2x+7\right)$$

To find equation solutions, solve -x+4=0 and 2x+7=0.

$$x=4 x=-\frac{7}{2}$$

Steps Using the Quadratic Formula

Subtract 3x^{2} from both sides.

$$x^{2}+2x+28-3x^{2}=x$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}+2x+28=x$$

Subtract x from both sides.

$$-2x^{2}+2x+28-x=0$$

Combine 2x and -x to get x.

$$-2x^{2}+x+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 1 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\times 28}}{2\left(-2\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-2\right)\times 28}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-1±\sqrt{1+8\times 28}}{2\left(-2\right)}$$

Multiply 8 times 28.

$$x=\frac{-1±\sqrt{1+224}}{2\left(-2\right)}$$

Add 1 to 224.

$$x=\frac{-1±\sqrt{225}}{2\left(-2\right)}$$

Take the square root of 225.

$$x=\frac{-1±15}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-1±15}{-4}$$

Now solve the equation x=\frac{-1±15}{-4} when ± is plus. Add -1 to 15.

$$x=\frac{14}{-4}$$

Reduce the fraction \frac{14}{-4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{7}{2}$$

Now solve the equation x=\frac{-1±15}{-4} when ± is minus. Subtract 15 from -1.

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

Divide -16 by -4.

$$x=4$$

The equation is now solved.

$$x=-\frac{7}{2} x=4$$

Steps for Completing the Square

Subtract 3x^{2} from both sides.

$$x^{2}+2x+28-3x^{2}=x$$

Combine x^{2} and -3x^{2} to get -2x^{2}.

$$-2x^{2}+2x+28=x$$

Subtract x from both sides.

$$-2x^{2}+2x+28-x=0$$

Combine 2x and -x to get x.

$$-2x^{2}+x+28=0$$

Subtract 28 from both sides. Anything subtracted from zero gives its negation.

$$-2x^{2}+x=-28$$

Divide both sides by -2.

$$\frac{-2x^{2}+x}{-2}=-\frac{28}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{1}{-2}x=-\frac{28}{-2}$$

Divide 1 by -2.

$$x^{2}-\frac{1}{2}x=-\frac{28}{-2}$$

Divide -28 by -2.

$$x^{2}-\frac{1}{2}x=14$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=14+\left(-\frac{1}{4}\right)^{2}$$

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=14+\frac{1}{16}$$

Add 14 to \frac{1}{16}.

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

Factor x^{2}-\frac{1}{2}x+\frac{1}{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}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Simplify.

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

Add \frac{1}{4} to both sides of the equation.

$$x=4 x=-\frac{7}{2}$$