Solve 3x(x-5)+16=-x

$$3x(x-5)+16=-x$$

Solve for x

$$x=2$$

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Steps Using Factoring By Grouping

Use the distributive property to multiply 3x by x-5.

$$3x^{2}-15x+16=-x$$

Add x to both sides.

$$3x^{2}-15x+16+x=0$$

Combine -15x and x to get -14x.

$$3x^{2}-14x+16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 3x^{2}+ax+bx+16. To find a and b, set up a system to be solved.

$$a+b=-14 ab=3\times 16=48$$

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 48.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Calculate the sum for each pair.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

The solution is the pair that gives sum -14.

$$a=-8 b=-6$$

Rewrite 3x^{2}-14x+16 as \left(3x^{2}-8x\right)+\left(-6x+16\right).

$$\left(3x^{2}-8x\right)+\left(-6x+16\right)$$

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

$$x\left(3x-8\right)-2\left(3x-8\right)$$

Factor out common term 3x-8 by using distributive property.

$$\left(3x-8\right)\left(x-2\right)$$

To find equation solutions, solve 3x-8=0 and x-2=0.

$$x=\frac{8}{3} x=2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by x-5.

$$3x^{2}-15x+16=-x$$

Add x to both sides.

$$3x^{2}-15x+16+x=0$$

Combine -15x and x to get -14x.

$$3x^{2}-14x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -14 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 3\times 16}}{2\times 3}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 3\times 16}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-14\right)±\sqrt{196-12\times 16}}{2\times 3}$$

Multiply -12 times 16.

$$x=\frac{-\left(-14\right)±\sqrt{196-192}}{2\times 3}$$

Add 196 to -192.

$$x=\frac{-\left(-14\right)±\sqrt{4}}{2\times 3}$$

Take the square root of 4.

$$x=\frac{-\left(-14\right)±2}{2\times 3}$$

The opposite of -14 is 14.

$$x=\frac{14±2}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{14±2}{6}$$

Now solve the equation x=\frac{14±2}{6} when ± is plus. Add 14 to 2.

$$x=\frac{16}{6}$$

Reduce the fraction \frac{16}{6} to lowest terms by extracting and canceling out 2.

$$x=\frac{8}{3}$$

Now solve the equation x=\frac{14±2}{6} when ± is minus. Subtract 2 from 14.

$$x=\frac{12}{6}$$

Divide 12 by 6.

$$x=2$$

The equation is now solved.

$$x=\frac{8}{3} x=2$$

Steps for Completing the Square

Use the distributive property to multiply 3x by x-5.

$$3x^{2}-15x+16=-x$$

Add x to both sides.

$$3x^{2}-15x+16+x=0$$

Combine -15x and x to get -14x.

$$3x^{2}-14x+16=0$$

Subtract 16 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}-14x=-16$$

Divide both sides by 3.

$$\frac{3x^{2}-14x}{3}=-\frac{16}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{14}{3}x=-\frac{16}{3}$$

Divide -\frac{14}{3}, the coefficient of the x term, by 2 to get -\frac{7}{3}. Then add the square of -\frac{7}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{14}{3}x+\left(-\frac{7}{3}\right)^{2}=-\frac{16}{3}+\left(-\frac{7}{3}\right)^{2}$$

Square -\frac{7}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=-\frac{16}{3}+\frac{49}{9}$$

Add -\frac{16}{3} to \frac{49}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=\frac{1}{9}$$

Factor x^{2}-\frac{14}{3}x+\frac{49}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{3}\right)^{2}=\frac{1}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Simplify.

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

Add \frac{7}{3} to both sides of the equation.

$$x=\frac{8}{3} x=2$$