Solve (x-1)(x+4)/5+x+1/2=1

$$\frac { ( x - 1 ) ( x + 4 ) } { 5 } + \frac { x + 1 } { 2 } = 1$$

Solve for x

$$x = -\frac{13}{2} = -6\frac{1}{2} = -6.5$$

$$x=1$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 10, the least common multiple of 5,2.

$$2\left(x-1\right)\left(x+4\right)+5\left(x+1\right)=10$$

Use the distributive property to multiply 2 by x-1.

$$\left(2x-2\right)\left(x+4\right)+5\left(x+1\right)=10$$

Use the distributive property to multiply 2x-2 by x+4 and combine like terms.

$$2x^{2}+6x-8+5\left(x+1\right)=10$$

Use the distributive property to multiply 5 by x+1.

$$2x^{2}+6x-8+5x+5=10$$

Combine 6x and 5x to get 11x.

$$2x^{2}+11x-8+5=10$$

Add -8 and 5 to get -3.

$$2x^{2}+11x-3=10$$

Subtract 10 from both sides.

$$2x^{2}+11x-3-10=0$$

Subtract 10 from -3 to get -13.

$$2x^{2}+11x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 11 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 2\left(-13\right)}}{2\times 2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 2\left(-13\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-11±\sqrt{121-8\left(-13\right)}}{2\times 2}$$

Multiply -8 times -13.

$$x=\frac{-11±\sqrt{121+104}}{2\times 2}$$

Add 121 to 104.

$$x=\frac{-11±\sqrt{225}}{2\times 2}$$

Take the square root of 225.

$$x=\frac{-11±15}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-11±15}{4}$$

Now solve the equation x=\frac{-11±15}{4} when ± is plus. Add -11 to 15.

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

Divide 4 by 4.

$$x=1$$

Now solve the equation x=\frac{-11±15}{4} when ± is minus. Subtract 15 from -11.

$$x=-\frac{26}{4}$$

Reduce the fraction \frac{-26}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{13}{2}$$

The equation is now solved.

$$x=1 x=-\frac{13}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 10, the least common multiple of 5,2.

$$2\left(x-1\right)\left(x+4\right)+5\left(x+1\right)=10$$

Use the distributive property to multiply 2 by x-1.

$$\left(2x-2\right)\left(x+4\right)+5\left(x+1\right)=10$$

Use the distributive property to multiply 2x-2 by x+4 and combine like terms.

$$2x^{2}+6x-8+5\left(x+1\right)=10$$

Use the distributive property to multiply 5 by x+1.

$$2x^{2}+6x-8+5x+5=10$$

Combine 6x and 5x to get 11x.

$$2x^{2}+11x-8+5=10$$

Add -8 and 5 to get -3.

$$2x^{2}+11x-3=10$$

Add 3 to both sides.

$$2x^{2}+11x=10+3$$

Add 10 and 3 to get 13.

$$2x^{2}+11x=13$$

Divide both sides by 2.

$$\frac{2x^{2}+11x}{2}=\frac{13}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{11}{2}x=\frac{13}{2}$$

Divide \frac{11}{2}, the coefficient of the x term, by 2 to get \frac{11}{4}. Then add the square of \frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{2}x+\left(\frac{11}{4}\right)^{2}=\frac{13}{2}+\left(\frac{11}{4}\right)^{2}$$

Square \frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=\frac{13}{2}+\frac{121}{16}$$

Add \frac{13}{2} to \frac{121}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=\frac{225}{16}$$

Factor x^{2}+\frac{11}{2}x+\frac{121}{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}.

$$\left(x+\frac{11}{4}\right)^{2}=\frac{225}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Simplify.

$$x+\frac{11}{4}=\frac{15}{4} x+\frac{11}{4}=-\frac{15}{4}$$

Subtract \frac{11}{4} from both sides of the equation.

$$x=1 x=-\frac{13}{2}$$