Solve 18-4.5x=64-32x+4x^2

$$18-4.5x=64-32x+4 { x }^{ 2 }$$

Solve for x

$$x=4$$

$$x=2.875$$

Steps Using the Quadratic Formula

Subtract 64 from both sides.

$$18-4.5x-64=-32x+4x^{2}$$

Subtract 64 from 18 to get -46.

$$-46-4.5x=-32x+4x^{2}$$

Add 32x to both sides.

$$-46-4.5x+32x=4x^{2}$$

Combine -4.5x and 32x to get 27.5x.

$$-46+27.5x=4x^{2}$$

Subtract 4x^{2} from both sides.

$$-46+27.5x-4x^{2}=0$$

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.

$$-4x^{2}+27.5x-46=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 27.5 for b, and -46 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27.5±\sqrt{27.5^{2}-4\left(-4\right)\left(-46\right)}}{2\left(-4\right)}$$

Square 27.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-27.5±\sqrt{756.25-4\left(-4\right)\left(-46\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-27.5±\sqrt{756.25+16\left(-46\right)}}{2\left(-4\right)}$$

Multiply 16 times -46.

$$x=\frac{-27.5±\sqrt{756.25-736}}{2\left(-4\right)}$$

Add 756.25 to -736.

$$x=\frac{-27.5±\sqrt{20.25}}{2\left(-4\right)}$$

Take the square root of 20.25.

$$x=\frac{-27.5±\frac{9}{2}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-27.5±\frac{9}{2}}{-8}$$

Now solve the equation x=\frac{-27.5±\frac{9}{2}}{-8} when ± is plus. Add -27.5 to \frac{9}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{23}{-8}$$

Divide -23 by -8.

$$x=\frac{23}{8}$$

Now solve the equation x=\frac{-27.5±\frac{9}{2}}{-8} when ± is minus. Subtract \frac{9}{2} from -27.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{32}{-8}$$

Divide -32 by -8.

$$x=4$$

The equation is now solved.

$$x=\frac{23}{8} x=4$$

Steps for Completing the Square

Add 32x to both sides.

$$18-4.5x+32x=64+4x^{2}$$

Combine -4.5x and 32x to get 27.5x.

$$18+27.5x=64+4x^{2}$$

Subtract 4x^{2} from both sides.

$$18+27.5x-4x^{2}=64$$

Subtract 18 from both sides.

$$27.5x-4x^{2}=64-18$$

Subtract 18 from 64 to get 46.

$$27.5x-4x^{2}=46$$

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.

$$-4x^{2}+27.5x=46$$

Divide both sides by -4.

$$\frac{-4x^{2}+27.5x}{-4}=\frac{46}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{27.5}{-4}x=\frac{46}{-4}$$

Divide 27.5 by -4.

$$x^{2}-6.875x=\frac{46}{-4}$$

Reduce the fraction \frac{46}{-4} to lowest terms by extracting and canceling out 2.

$$x^{2}-6.875x=-\frac{23}{2}$$

Divide -6.875, the coefficient of the x term, by 2 to get -3.4375. Then add the square of -3.4375 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6.875x+\left(-3.4375\right)^{2}=-\frac{23}{2}+\left(-3.4375\right)^{2}$$

Square -3.4375 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-6.875x+11.81640625=-\frac{23}{2}+11.81640625$$

Add -\frac{23}{2} to 11.81640625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-6.875x+11.81640625=\frac{81}{256}$$

Factor x^{2}-6.875x+11.81640625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3.4375\right)^{2}=\frac{81}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3.4375\right)^{2}}=\sqrt{\frac{81}{256}}$$

Simplify.

$$x-3.4375=\frac{9}{16} x-3.4375=-\frac{9}{16}$$

Add 3.4375 to both sides of the equation.

$$x=4 x=\frac{23}{8}$$