Solve 4(x+1)0.125+(x*(x+1))0.25=68

$$4 ( x + 1 ) 0.125 + ( x \cdot ( x + 1 ) ) 0.25 = 68$$

Solve for x

$$x=-18$$

$$x=15$$

Steps Using the Quadratic Formula

Multiply 4 and 0.125 to get 0.5.

$$0.5\left(x+1\right)+x\left(x+1\right)\times 0.25=68$$

Use the distributive property to multiply 0.5 by x+1.

$$0.5x+0.5+x\left(x+1\right)\times 0.25=68$$

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

$$0.5x+0.5+\left(x^{2}+x\right)\times 0.25=68$$

Use the distributive property to multiply x^{2}+x by 0.25.

$$0.5x+0.5+0.25x^{2}+0.25x=68$$

Combine 0.5x and 0.25x to get 0.75x.

$$0.75x+0.5+0.25x^{2}=68$$

Subtract 68 from both sides.

$$0.75x+0.5+0.25x^{2}-68=0$$

Subtract 68 from 0.5 to get -67.5.

$$0.75x-67.5+0.25x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$0.25x^{2}+0.75x-67.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.25 for a, 0.75 for b, and -67.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.75±\sqrt{0.75^{2}-4\times 0.25\left(-67.5\right)}}{2\times 0.25}$$

Square 0.75 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.75±\sqrt{0.5625-4\times 0.25\left(-67.5\right)}}{2\times 0.25}$$

Multiply -4 times 0.25.

$$x=\frac{-0.75±\sqrt{0.5625-\left(-67.5\right)}}{2\times 0.25}$$

Multiply -1 times -67.5.

$$x=\frac{-0.75±\sqrt{0.5625+67.5}}{2\times 0.25}$$

Add 0.5625 to 67.5 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.75±\sqrt{68.0625}}{2\times 0.25}$$

Take the square root of 68.0625.

$$x=\frac{-0.75±\frac{33}{4}}{2\times 0.25}$$

Multiply 2 times 0.25.

$$x=\frac{-0.75±\frac{33}{4}}{0.5}$$

Now solve the equation x=\frac{-0.75±\frac{33}{4}}{0.5} when ± is plus. Add -0.75 to \frac{33}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{15}{2}}{0.5}$$

Divide \frac{15}{2} by 0.5 by multiplying \frac{15}{2} by the reciprocal of 0.5.

$$x=15$$

Now solve the equation x=\frac{-0.75±\frac{33}{4}}{0.5} when ± is minus. Subtract \frac{33}{4} from -0.75 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{9}{0.5}$$

Divide -9 by 0.5 by multiplying -9 by the reciprocal of 0.5.

$$x=-18$$

The equation is now solved.

$$x=15 x=-18$$

Steps for Completing the Square

Multiply 4 and 0.125 to get 0.5.

$$0.5\left(x+1\right)+x\left(x+1\right)\times 0.25=68$$

Use the distributive property to multiply 0.5 by x+1.

$$0.5x+0.5+x\left(x+1\right)\times 0.25=68$$

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

$$0.5x+0.5+\left(x^{2}+x\right)\times 0.25=68$$

Use the distributive property to multiply x^{2}+x by 0.25.

$$0.5x+0.5+0.25x^{2}+0.25x=68$$

Combine 0.5x and 0.25x to get 0.75x.

$$0.75x+0.5+0.25x^{2}=68$$

Subtract 0.5 from both sides.

$$0.75x+0.25x^{2}=68-0.5$$

Subtract 0.5 from 68 to get 67.5.

$$0.75x+0.25x^{2}=67.5$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$0.25x^{2}+0.75x=67.5$$

Multiply both sides by 4.

$$\frac{0.25x^{2}+0.75x}{0.25}=\frac{67.5}{0.25}$$

Dividing by 0.25 undoes the multiplication by 0.25.

$$x^{2}+\frac{0.75}{0.25}x=\frac{67.5}{0.25}$$

Divide 0.75 by 0.25 by multiplying 0.75 by the reciprocal of 0.25.

$$x^{2}+3x=\frac{67.5}{0.25}$$

Divide 67.5 by 0.25 by multiplying 67.5 by the reciprocal of 0.25.

$$x^{2}+3x=270$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=270+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=270+\frac{9}{4}$$

Add 270 to \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{1089}{4}$$

Factor x^{2}+3x+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{1089}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{1089}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{33}{2} x+\frac{3}{2}=-\frac{33}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=15 x=-18$$