Solve 7x^2+27x-2=4x-8

$$7 x ^ { 2 } + 27 x - 2 = 4 x - 8$$

Solve for x

$$x=-3$$

$$x=-\frac{2}{7}\approx -0.285714286$$

Steps Using Factoring By Grouping

Subtract 4x from both sides.

$$7x^{2}+27x-2-4x=-8$$

Combine 27x and -4x to get 23x.

$$7x^{2}+23x-2=-8$$

Add 8 to both sides.

$$7x^{2}+23x-2+8=0$$

Add -2 and 8 to get 6.

$$7x^{2}+23x+6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=23 ab=7\times 6=42$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 42.

$$1,42 2,21 3,14 6,7$$

Calculate the sum for each pair.

$$1+42=43 2+21=23 3+14=17 6+7=13$$

The solution is the pair that gives sum 23.

$$a=2 b=21$$

Rewrite 7x^{2}+23x+6 as \left(7x^{2}+2x\right)+\left(21x+6\right).

$$\left(7x^{2}+2x\right)+\left(21x+6\right)$$

Factor out x in the first and 3 in the second group.

$$x\left(7x+2\right)+3\left(7x+2\right)$$

Factor out common term 7x+2 by using distributive property.

$$\left(7x+2\right)\left(x+3\right)$$

To find equation solutions, solve 7x+2=0 and x+3=0.

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

Steps Using the Quadratic Formula

Subtract 4x from both sides.

$$7x^{2}+27x-2-4x=-8$$

Combine 27x and -4x to get 23x.

$$7x^{2}+23x-2=-8$$

Add 8 to both sides.

$$7x^{2}+23x-2+8=0$$

Add -2 and 8 to get 6.

$$7x^{2}+23x+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 23 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-23±\sqrt{23^{2}-4\times 7\times 6}}{2\times 7}$$

Square 23.

$$x=\frac{-23±\sqrt{529-4\times 7\times 6}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-23±\sqrt{529-28\times 6}}{2\times 7}$$

Multiply -28 times 6.

$$x=\frac{-23±\sqrt{529-168}}{2\times 7}$$

Add 529 to -168.

$$x=\frac{-23±\sqrt{361}}{2\times 7}$$

Take the square root of 361.

$$x=\frac{-23±19}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-23±19}{14}$$

Now solve the equation x=\frac{-23±19}{14} when ± is plus. Add -23 to 19.

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

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

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

Now solve the equation x=\frac{-23±19}{14} when ± is minus. Subtract 19 from -23.

$$x=-\frac{42}{14}$$

Divide -42 by 14.

$$x=-3$$

The equation is now solved.

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

Steps for Completing the Square

Subtract 4x from both sides.

$$7x^{2}+27x-2-4x=-8$$

Combine 27x and -4x to get 23x.

$$7x^{2}+23x-2=-8$$

Add 2 to both sides.

$$7x^{2}+23x=-8+2$$

Add -8 and 2 to get -6.

$$7x^{2}+23x=-6$$

Divide both sides by 7.

$$\frac{7x^{2}+23x}{7}=-\frac{6}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\frac{23}{7}x=-\frac{6}{7}$$

Divide \frac{23}{7}, the coefficient of the x term, by 2 to get \frac{23}{14}. Then add the square of \frac{23}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{23}{7}x+\left(\frac{23}{14}\right)^{2}=-\frac{6}{7}+\left(\frac{23}{14}\right)^{2}$$

Square \frac{23}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{23}{7}x+\frac{529}{196}=-\frac{6}{7}+\frac{529}{196}$$

Add -\frac{6}{7} to \frac{529}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{23}{7}x+\frac{529}{196}=\frac{361}{196}$$

Factor x^{2}+\frac{23}{7}x+\frac{529}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{23}{14}\right)^{2}=\frac{361}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{23}{14}\right)^{2}}=\sqrt{\frac{361}{196}}$$

Simplify.

$$x+\frac{23}{14}=\frac{19}{14} x+\frac{23}{14}=-\frac{19}{14}$$

Subtract \frac{23}{14} from both sides of the equation.

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