Solve 5x^2+7x-6/x+2=x+4

$$\frac { 5 x ^ { 2 } + 7 x - 6 } { x + 2 } = x + 4$$

Solve for x

$$x = \frac{7}{4} = 1\frac{3}{4} = 1.75$$

Steps Using Factoring By Grouping

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$5x^{2}+7x-6=\left(x+2\right)x+\left(x+2\right)\times 4$$

Use the distributive property to multiply x+2 by x.

$$5x^{2}+7x-6=x^{2}+2x+\left(x+2\right)\times 4$$

Use the distributive property to multiply x+2 by 4.

$$5x^{2}+7x-6=x^{2}+2x+4x+8$$

Combine 2x and 4x to get 6x.

$$5x^{2}+7x-6=x^{2}+6x+8$$

Subtract x^{2} from both sides.

$$5x^{2}+7x-6-x^{2}=6x+8$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}+7x-6=6x+8$$

Subtract 6x from both sides.

$$4x^{2}+7x-6-6x=8$$

Combine 7x and -6x to get x.

$$4x^{2}+x-6=8$$

Subtract 8 from both sides.

$$4x^{2}+x-6-8=0$$

Subtract 8 from -6 to get -14.

$$4x^{2}+x-14=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4x^{2}+ax+bx-14. To find a and b, set up a system to be solved.

$$a+b=1 ab=4\left(-14\right)=-56$$

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.

$$a=-7 b=8$$

Rewrite 4x^{2}+x-14 as \left(4x^{2}-7x\right)+\left(8x-14\right).

$$\left(4x^{2}-7x\right)+\left(8x-14\right)$$

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

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

Factor out common term 4x-7 by using distributive property.

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

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

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

Variable x cannot be equal to -2.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$5x^{2}+7x-6=\left(x+2\right)x+\left(x+2\right)\times 4$$

Use the distributive property to multiply x+2 by x.

$$5x^{2}+7x-6=x^{2}+2x+\left(x+2\right)\times 4$$

Use the distributive property to multiply x+2 by 4.

$$5x^{2}+7x-6=x^{2}+2x+4x+8$$

Combine 2x and 4x to get 6x.

$$5x^{2}+7x-6=x^{2}+6x+8$$

Subtract x^{2} from both sides.

$$5x^{2}+7x-6-x^{2}=6x+8$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}+7x-6=6x+8$$

Subtract 6x from both sides.

$$4x^{2}+7x-6-6x=8$$

Combine 7x and -6x to get x.

$$4x^{2}+x-6=8$$

Subtract 8 from both sides.

$$4x^{2}+x-6-8=0$$

Subtract 8 from -6 to get -14.

$$4x^{2}+x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 1 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 4\left(-14\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-1±\sqrt{1-16\left(-14\right)}}{2\times 4}$$

Multiply -16 times -14.

$$x=\frac{-1±\sqrt{1+224}}{2\times 4}$$

Add 1 to 224.

$$x=\frac{-1±\sqrt{225}}{2\times 4}$$

Take the square root of 225.

$$x=\frac{-1±15}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-1±15}{8}$$

Now solve the equation x=\frac{-1±15}{8} when ± is plus. Add -1 to 15.

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

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

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

Now solve the equation x=\frac{-1±15}{8} when ± is minus. Subtract 15 from -1.

$$x=-\frac{16}{8}$$

Divide -16 by 8.

$$x=-2$$

The equation is now solved.

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

Variable x cannot be equal to -2.

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

Steps for Completing the Square

Variable x cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by x+2.

$$5x^{2}+7x-6=\left(x+2\right)x+\left(x+2\right)\times 4$$

Use the distributive property to multiply x+2 by x.

$$5x^{2}+7x-6=x^{2}+2x+\left(x+2\right)\times 4$$

Use the distributive property to multiply x+2 by 4.

$$5x^{2}+7x-6=x^{2}+2x+4x+8$$

Combine 2x and 4x to get 6x.

$$5x^{2}+7x-6=x^{2}+6x+8$$

Subtract x^{2} from both sides.

$$5x^{2}+7x-6-x^{2}=6x+8$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}+7x-6=6x+8$$

Subtract 6x from both sides.

$$4x^{2}+7x-6-6x=8$$

Combine 7x and -6x to get x.

$$4x^{2}+x-6=8$$

Add 6 to both sides.

$$4x^{2}+x=8+6$$

Add 8 and 6 to get 14.

$$4x^{2}+x=14$$

Divide both sides by 4.

$$\frac{4x^{2}+x}{4}=\frac{14}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{1}{4}x=\frac{14}{4}$$

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

$$x^{2}+\frac{1}{4}x=\frac{7}{2}$$

Divide \frac{1}{4}, the coefficient of the x term, by 2 to get \frac{1}{8}. Then add the square of \frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{4}x+\left(\frac{1}{8}\right)^{2}=\frac{7}{2}+\left(\frac{1}{8}\right)^{2}$$

Square \frac{1}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{7}{2}+\frac{1}{64}$$

Add \frac{7}{2} to \frac{1}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{225}{64}$$

Factor x^{2}+\frac{1}{4}x+\frac{1}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{8}\right)^{2}=\frac{225}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{8}\right)^{2}}=\sqrt{\frac{225}{64}}$$

Simplify.

$$x+\frac{1}{8}=\frac{15}{8} x+\frac{1}{8}=-\frac{15}{8}$$

Subtract \frac{1}{8} from both sides of the equation.

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

Variable x cannot be equal to -2.

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