Solve 7x^2-10x-8=(7x+4)

$$7 x ^ { 2 } - 10 x - 8 = ( 7 x + 4 )$$

Solve for x

$$x=-\frac{4}{7}\approx -0.571428571$$

$$x=3$$

Steps Using Factoring By Grouping

Subtract 7x from both sides.

$$7x^{2}-10x-8-7x=4$$

Combine -10x and -7x to get -17x.

$$7x^{2}-17x-8=4$$

Subtract 4 from both sides.

$$7x^{2}-17x-8-4=0$$

Subtract 4 from -8 to get -12.

$$7x^{2}-17x-12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 7x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=-17 ab=7\left(-12\right)=-84$$

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 -84.

$$1,-84 2,-42 3,-28 4,-21 6,-14 7,-12$$

Calculate the sum for each pair.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

The solution is the pair that gives sum -17.

$$a=-21 b=4$$

Rewrite 7x^{2}-17x-12 as \left(7x^{2}-21x\right)+\left(4x-12\right).

$$\left(7x^{2}-21x\right)+\left(4x-12\right)$$

Factor out 7x in the first and 4 in the second group.

$$7x\left(x-3\right)+4\left(x-3\right)$$

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

$$\left(x-3\right)\left(7x+4\right)$$

To find equation solutions, solve x-3=0 and 7x+4=0.

$$x=3 x=-\frac{4}{7}$$

Steps Using the Quadratic Formula

Subtract 7x from both sides.

$$7x^{2}-10x-8-7x=4$$

Combine -10x and -7x to get -17x.

$$7x^{2}-17x-8=4$$

Subtract 4 from both sides.

$$7x^{2}-17x-8-4=0$$

Subtract 4 from -8 to get -12.

$$7x^{2}-17x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -17 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 7\left(-12\right)}}{2\times 7}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 7\left(-12\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-17\right)±\sqrt{289-28\left(-12\right)}}{2\times 7}$$

Multiply -28 times -12.

$$x=\frac{-\left(-17\right)±\sqrt{289+336}}{2\times 7}$$

Add 289 to 336.

$$x=\frac{-\left(-17\right)±\sqrt{625}}{2\times 7}$$

Take the square root of 625.

$$x=\frac{-\left(-17\right)±25}{2\times 7}$$

The opposite of -17 is 17.

$$x=\frac{17±25}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{17±25}{14}$$

Now solve the equation x=\frac{17±25}{14} when ± is plus. Add 17 to 25.

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

Divide 42 by 14.

$$x=3$$

Now solve the equation x=\frac{17±25}{14} when ± is minus. Subtract 25 from 17.

$$x=-\frac{8}{14}$$

Reduce the fraction \frac{-8}{14} to lowest terms by extracting and canceling out 2.

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

The equation is now solved.

$$x=3 x=-\frac{4}{7}$$

Steps for Completing the Square

Subtract 7x from both sides.

$$7x^{2}-10x-8-7x=4$$

Combine -10x and -7x to get -17x.

$$7x^{2}-17x-8=4$$

Add 8 to both sides.

$$7x^{2}-17x=4+8$$

Add 4 and 8 to get 12.

$$7x^{2}-17x=12$$

Divide both sides by 7.

$$\frac{7x^{2}-17x}{7}=\frac{12}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{17}{7}x=\frac{12}{7}$$

Divide -\frac{17}{7}, the coefficient of the x term, by 2 to get -\frac{17}{14}. Then add the square of -\frac{17}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{17}{7}x+\left(-\frac{17}{14}\right)^{2}=\frac{12}{7}+\left(-\frac{17}{14}\right)^{2}$$

Square -\frac{17}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{17}{7}x+\frac{289}{196}=\frac{12}{7}+\frac{289}{196}$$

Add \frac{12}{7} to \frac{289}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{17}{7}x+\frac{289}{196}=\frac{625}{196}$$

Factor x^{2}-\frac{17}{7}x+\frac{289}{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}.

$$\left(x-\frac{17}{14}\right)^{2}=\frac{625}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{17}{14}\right)^{2}}=\sqrt{\frac{625}{196}}$$

Simplify.

$$x-\frac{17}{14}=\frac{25}{14} x-\frac{17}{14}=-\frac{25}{14}$$

Add \frac{17}{14} to both sides of the equation.

$$x=3 x=-\frac{4}{7}$$