Solve 160x+(150-2x)*x=150*80*1/8

$$160x+(150-2x) \times x=150 \times 80 \times \frac{ 1 }{ 8 }$$

Solve for x

$$x=5$$

$$x=150$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 150-2x by x.

$$160x+150x-2x^{2}=150\times 80\times \frac{1}{8}$$

Combine 160x and 150x to get 310x.

$$310x-2x^{2}=150\times 80\times \frac{1}{8}$$

Multiply 150 and 80 to get 12000.

$$310x-2x^{2}=12000\times \frac{1}{8}$$

Multiply 12000 and \frac{1}{8} to get \frac{12000}{8}.

$$310x-2x^{2}=\frac{12000}{8}$$

Divide 12000 by 8 to get 1500.

$$310x-2x^{2}=1500$$

Subtract 1500 from both sides.

$$310x-2x^{2}-1500=0$$

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.

$$-2x^{2}+310x-1500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 310 for b, and -1500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-310±\sqrt{310^{2}-4\left(-2\right)\left(-1500\right)}}{2\left(-2\right)}$$

Square 310.

$$x=\frac{-310±\sqrt{96100-4\left(-2\right)\left(-1500\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-310±\sqrt{96100+8\left(-1500\right)}}{2\left(-2\right)}$$

Multiply 8 times -1500.

$$x=\frac{-310±\sqrt{96100-12000}}{2\left(-2\right)}$$

Add 96100 to -12000.

$$x=\frac{-310±\sqrt{84100}}{2\left(-2\right)}$$

Take the square root of 84100.

$$x=\frac{-310±290}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-310±290}{-4}$$

Now solve the equation x=\frac{-310±290}{-4} when ± is plus. Add -310 to 290.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

Now solve the equation x=\frac{-310±290}{-4} when ± is minus. Subtract 290 from -310.

$$x=-\frac{600}{-4}$$

Divide -600 by -4.

$$x=150$$

The equation is now solved.

$$x=5 x=150$$

Steps for Completing the Square

Use the distributive property to multiply 150-2x by x.

$$160x+150x-2x^{2}=150\times 80\times \frac{1}{8}$$

Combine 160x and 150x to get 310x.

$$310x-2x^{2}=150\times 80\times \frac{1}{8}$$

Multiply 150 and 80 to get 12000.

$$310x-2x^{2}=12000\times \frac{1}{8}$$

Multiply 12000 and \frac{1}{8} to get \frac{12000}{8}.

$$310x-2x^{2}=\frac{12000}{8}$$

Divide 12000 by 8 to get 1500.

$$310x-2x^{2}=1500$$

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.

$$-2x^{2}+310x=1500$$

Divide both sides by -2.

$$\frac{-2x^{2}+310x}{-2}=\frac{1500}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{310}{-2}x=\frac{1500}{-2}$$

Divide 310 by -2.

$$x^{2}-155x=\frac{1500}{-2}$$

Divide 1500 by -2.

$$x^{2}-155x=-750$$

Divide -155, the coefficient of the x term, by 2 to get -\frac{155}{2}. Then add the square of -\frac{155}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-155x+\left(-\frac{155}{2}\right)^{2}=-750+\left(-\frac{155}{2}\right)^{2}$$

Square -\frac{155}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-155x+\frac{24025}{4}=-750+\frac{24025}{4}$$

Add -750 to \frac{24025}{4}.

$$x^{2}-155x+\frac{24025}{4}=\frac{21025}{4}$$

Factor x^{2}-155x+\frac{24025}{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}.

$$\left(x-\frac{155}{2}\right)^{2}=\frac{21025}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{155}{2}\right)^{2}}=\sqrt{\frac{21025}{4}}$$

Simplify.

$$x-\frac{155}{2}=\frac{145}{2} x-\frac{155}{2}=-\frac{145}{2}$$

Add \frac{155}{2} to both sides of the equation.

$$x=150 x=5$$