Solve (40-2x)*x+52x=176

$$(40-2x) \times x+52x=176$$

Solve for x

$$x=2$$

$$x=44$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40-2x by x.

$$40x-2x^{2}+52x=176$$

Combine 40x and 52x to get 92x.

$$92x-2x^{2}=176$$

Subtract 176 from both sides.

$$92x-2x^{2}-176=0$$

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.

$$-2x^{2}+92x-176=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 92 for b, and -176 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-92±\sqrt{92^{2}-4\left(-2\right)\left(-176\right)}}{2\left(-2\right)}$$

Square 92.

$$x=\frac{-92±\sqrt{8464-4\left(-2\right)\left(-176\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-92±\sqrt{8464+8\left(-176\right)}}{2\left(-2\right)}$$

Multiply 8 times -176.

$$x=\frac{-92±\sqrt{8464-1408}}{2\left(-2\right)}$$

Add 8464 to -1408.

$$x=\frac{-92±\sqrt{7056}}{2\left(-2\right)}$$

Take the square root of 7056.

$$x=\frac{-92±84}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-92±84}{-4}$$

Now solve the equation x=\frac{-92±84}{-4} when ± is plus. Add -92 to 84.

$$x=-\frac{8}{-4}$$

Divide -8 by -4.

$$x=2$$

Now solve the equation x=\frac{-92±84}{-4} when ± is minus. Subtract 84 from -92.

$$x=-\frac{176}{-4}$$

Divide -176 by -4.

$$x=44$$

The equation is now solved.

$$x=2 x=44$$

Steps for Completing the Square

Use the distributive property to multiply 40-2x by x.

$$40x-2x^{2}+52x=176$$

Combine 40x and 52x to get 92x.

$$92x-2x^{2}=176$$

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.

$$-2x^{2}+92x=176$$

Divide both sides by -2.

$$\frac{-2x^{2}+92x}{-2}=\frac{176}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{92}{-2}x=\frac{176}{-2}$$

Divide 92 by -2.

$$x^{2}-46x=\frac{176}{-2}$$

Divide 176 by -2.

$$x^{2}-46x=-88$$

Divide -46, the coefficient of the x term, by 2 to get -23. Then add the square of -23 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-46x+\left(-23\right)^{2}=-88+\left(-23\right)^{2}$$

Square -23.

$$x^{2}-46x+529=-88+529$$

Add -88 to 529.

$$x^{2}-46x+529=441$$

Factor x^{2}-46x+529. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-23\right)^{2}=441$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-23\right)^{2}}=\sqrt{441}$$

Simplify.

$$x-23=21 x-23=-21$$

Add 23 to both sides of the equation.

$$x=44 x=2$$