Solve 0.5*(4+x)=2.25*4+x/4*x

$$0.5 \times (4+x)=2.25 \times \frac{ 4+x }{ 4 \times x }$$

Solve for x

$$x=-4$$

$$x=1.125$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x.

$$0.5\left(4+x\right)\times 4x=2.25\left(4+x\right)$$

Multiply 0.5 and 4 to get 2.

$$2\left(4+x\right)x=2.25\left(4+x\right)$$

Use the distributive property to multiply 2 by 4+x.

$$\left(8+2x\right)x=2.25\left(4+x\right)$$

Use the distributive property to multiply 8+2x by x.

$$8x+2x^{2}=2.25\left(4+x\right)$$

Use the distributive property to multiply 2.25 by 4+x.

$$8x+2x^{2}=9+2.25x$$

Subtract 9 from both sides.

$$8x+2x^{2}-9=2.25x$$

Subtract 2.25x from both sides.

$$8x+2x^{2}-9-2.25x=0$$

Combine 8x and -2.25x to get 5.75x.

$$5.75x+2x^{2}-9=0$$

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.

$$2x^{2}+5.75x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 5.75 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5.75±\sqrt{5.75^{2}-4\times 2\left(-9\right)}}{2\times 2}$$

Square 5.75 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-5.75±\sqrt{33.0625-4\times 2\left(-9\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5.75±\sqrt{33.0625-8\left(-9\right)}}{2\times 2}$$

Multiply -8 times -9.

$$x=\frac{-5.75±\sqrt{33.0625+72}}{2\times 2}$$

Add 33.0625 to 72.

$$x=\frac{-5.75±\sqrt{105.0625}}{2\times 2}$$

Take the square root of 105.0625.

$$x=\frac{-5.75±\frac{41}{4}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5.75±\frac{41}{4}}{4}$$

Now solve the equation x=\frac{-5.75±\frac{41}{4}}{4} when ± is plus. Add -5.75 to \frac{41}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{9}{2}}{4}$$

Divide \frac{9}{2} by 4.

$$x=\frac{9}{8}$$

Now solve the equation x=\frac{-5.75±\frac{41}{4}}{4} when ± is minus. Subtract \frac{41}{4} from -5.75 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{16}{4}$$

Divide -16 by 4.

$$x=-4$$

The equation is now solved.

$$x=\frac{9}{8} x=-4$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x.

$$0.5\left(4+x\right)\times 4x=2.25\left(4+x\right)$$

Multiply 0.5 and 4 to get 2.

$$2\left(4+x\right)x=2.25\left(4+x\right)$$

Use the distributive property to multiply 2 by 4+x.

$$\left(8+2x\right)x=2.25\left(4+x\right)$$

Use the distributive property to multiply 8+2x by x.

$$8x+2x^{2}=2.25\left(4+x\right)$$

Use the distributive property to multiply 2.25 by 4+x.

$$8x+2x^{2}=9+2.25x$$

Subtract 2.25x from both sides.

$$8x+2x^{2}-2.25x=9$$

Combine 8x and -2.25x to get 5.75x.

$$5.75x+2x^{2}=9$$

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.

$$2x^{2}+5.75x=9$$

Divide both sides by 2.

$$\frac{2x^{2}+5.75x}{2}=\frac{9}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{5.75}{2}x=\frac{9}{2}$$

Divide 5.75 by 2.

$$x^{2}+2.875x=\frac{9}{2}$$

Divide 2.875, the coefficient of the x term, by 2 to get 1.4375. Then add the square of 1.4375 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2.875x+1.4375^{2}=\frac{9}{2}+1.4375^{2}$$

Square 1.4375 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+2.875x+2.06640625=\frac{9}{2}+2.06640625$$

Add \frac{9}{2} to 2.06640625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+2.875x+2.06640625=\frac{1681}{256}$$

Factor x^{2}+2.875x+2.06640625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1.4375\right)^{2}=\frac{1681}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1.4375\right)^{2}}=\sqrt{\frac{1681}{256}}$$

Simplify.

$$x+1.4375=\frac{41}{16} x+1.4375=-\frac{41}{16}$$

Subtract 1.4375 from both sides of the equation.

$$x=\frac{9}{8} x=-4$$