Solve 375=x(4+(x-1)3)/2

$$375= \frac{ x(4+(x-1)3) }{ 2 }$$

Solve for x

$$x = \frac{\sqrt{9001} - 1}{6} \approx 15.64560002$$

$$x=\frac{-\sqrt{9001}-1}{6}\approx -15.978933354$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$375\times 2=x\left(4+\left(x-1\right)\times 3\right)$$

Multiply 375 and 2 to get 750.

$$750=x\left(4+\left(x-1\right)\times 3\right)$$

Use the distributive property to multiply x-1 by 3.

$$750=x\left(4+3x-3\right)$$

Subtract 3 from 4 to get 1.

$$750=x\left(1+3x\right)$$

Use the distributive property to multiply x by 1+3x.

$$750=x+3x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$x+3x^{2}=750$$

Subtract 750 from both sides.

$$x+3x^{2}-750=0$$

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.

$$3x^{2}+x-750=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 1 for b, and -750 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 3\left(-750\right)}}{2\times 3}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 3\left(-750\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-1±\sqrt{1-12\left(-750\right)}}{2\times 3}$$

Multiply -12 times -750.

$$x=\frac{-1±\sqrt{1+9000}}{2\times 3}$$

Add 1 to 9000.

$$x=\frac{-1±\sqrt{9001}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-1±\sqrt{9001}}{6}$$

Now solve the equation x=\frac{-1±\sqrt{9001}}{6} when ± is plus. Add -1 to \sqrt{9001}.

$$x=\frac{\sqrt{9001}-1}{6}$$

Now solve the equation x=\frac{-1±\sqrt{9001}}{6} when ± is minus. Subtract \sqrt{9001} from -1.

$$x=\frac{-\sqrt{9001}-1}{6}$$

The equation is now solved.

$$x=\frac{\sqrt{9001}-1}{6} x=\frac{-\sqrt{9001}-1}{6}$$

Steps for Completing the Square

Multiply both sides by 2.

$$375\times 2=x\left(4+\left(x-1\right)\times 3\right)$$

Multiply 375 and 2 to get 750.

$$750=x\left(4+\left(x-1\right)\times 3\right)$$

Use the distributive property to multiply x-1 by 3.

$$750=x\left(4+3x-3\right)$$

Subtract 3 from 4 to get 1.

$$750=x\left(1+3x\right)$$

Use the distributive property to multiply x by 1+3x.

$$750=x+3x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$x+3x^{2}=750$$

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.

$$3x^{2}+x=750$$

Divide both sides by 3.

$$\frac{3x^{2}+x}{3}=\frac{750}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{1}{3}x=\frac{750}{3}$$

Divide 750 by 3.

$$x^{2}+\frac{1}{3}x=250$$

Divide \frac{1}{3}, the coefficient of the x term, by 2 to get \frac{1}{6}. Then add the square of \frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{3}x+\left(\frac{1}{6}\right)^{2}=250+\left(\frac{1}{6}\right)^{2}$$

Square \frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=250+\frac{1}{36}$$

Add 250 to \frac{1}{36}.

$$x^{2}+\frac{1}{3}x+\frac{1}{36}=\frac{9001}{36}$$

Factor x^{2}+\frac{1}{3}x+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{6}\right)^{2}=\frac{9001}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{6}\right)^{2}}=\sqrt{\frac{9001}{36}}$$

Simplify.

$$x+\frac{1}{6}=\frac{\sqrt{9001}}{6} x+\frac{1}{6}=-\frac{\sqrt{9001}}{6}$$

Subtract \frac{1}{6} from both sides of the equation.

$$x=\frac{\sqrt{9001}-1}{6} x=\frac{-\sqrt{9001}-1}{6}$$