Solve 4,704=0.4x^2+0.5(0.2(2.4-x))^2

$$4,704 = 0.4 x ^ { 2 } + 0.5 ( 0.2 ( 2.4 - x ) ) ^ { 2 }$$

Solve for x

$$x = \frac{8 \sqrt{214370} + 4}{35} \approx 105.94310398$$

$$x=\frac{4-8\sqrt{214370}}{35}\approx -105.714532551$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.2 by 2.4-x.

$$4704=0.4x^{2}+0.5\left(0.48-0.2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.48-0.2x\right)^{2}.

$$4704=0.4x^{2}+0.5\left(0.2304-0.192x+0.04x^{2}\right)$$

Use the distributive property to multiply 0.5 by 0.2304-0.192x+0.04x^{2}.

$$4704=0.4x^{2}+0.1152-0.096x+0.02x^{2}$$

Combine 0.4x^{2} and 0.02x^{2} to get 0.42x^{2}.

$$4704=0.42x^{2}+0.1152-0.096x$$

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

$$0.42x^{2}+0.1152-0.096x=4704$$

Subtract 4704 from both sides.

$$0.42x^{2}+0.1152-0.096x-4704=0$$

Subtract 4704 from 0.1152 to get -4703.8848.

$$0.42x^{2}-4703.8848-0.096x=0$$

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.

$$0.42x^{2}-0.096x-4703.8848=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.42 for a, -0.096 for b, and -4703.8848 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.096\right)±\sqrt{\left(-0.096\right)^{2}-4\times 0.42\left(-4703.8848\right)}}{2\times 0.42}$$

Square -0.096 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.096\right)±\sqrt{0.009216-4\times 0.42\left(-4703.8848\right)}}{2\times 0.42}$$

Multiply -4 times 0.42.

$$x=\frac{-\left(-0.096\right)±\sqrt{0.009216-1.68\left(-4703.8848\right)}}{2\times 0.42}$$

Multiply -1.68 times -4703.8848 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.096\right)±\sqrt{\frac{144+123476976}{15625}}}{2\times 0.42}$$

Add 0.009216 to 7902.526464 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.096\right)±\sqrt{7902.53568}}{2\times 0.42}$$

Take the square root of 7902.53568.

$$x=\frac{-\left(-0.096\right)±\frac{24\sqrt{214370}}{125}}{2\times 0.42}$$

The opposite of -0.096 is 0.096.

$$x=\frac{0.096±\frac{24\sqrt{214370}}{125}}{2\times 0.42}$$

Multiply 2 times 0.42.

$$x=\frac{0.096±\frac{24\sqrt{214370}}{125}}{0.84}$$

Now solve the equation x=\frac{0.096±\frac{24\sqrt{214370}}{125}}{0.84} when ± is plus. Add 0.096 to \frac{24\sqrt{214370}}{125}.

$$x=\frac{24\sqrt{214370}+12}{0.84\times 125}$$

Divide \frac{12+24\sqrt{214370}}{125} by 0.84 by multiplying \frac{12+24\sqrt{214370}}{125} by the reciprocal of 0.84.

$$x=\frac{8\sqrt{214370}+4}{35}$$

Now solve the equation x=\frac{0.096±\frac{24\sqrt{214370}}{125}}{0.84} when ± is minus. Subtract \frac{24\sqrt{214370}}{125} from 0.096.

$$x=\frac{12-24\sqrt{214370}}{0.84\times 125}$$

Divide \frac{12-24\sqrt{214370}}{125} by 0.84 by multiplying \frac{12-24\sqrt{214370}}{125} by the reciprocal of 0.84.

$$x=\frac{4-8\sqrt{214370}}{35}$$

The equation is now solved.

$$x=\frac{8\sqrt{214370}+4}{35} x=\frac{4-8\sqrt{214370}}{35}$$

Steps for Completing the Square

Use the distributive property to multiply 0.2 by 2.4-x.

$$4704=0.4x^{2}+0.5\left(0.48-0.2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.48-0.2x\right)^{2}.

$$4704=0.4x^{2}+0.5\left(0.2304-0.192x+0.04x^{2}\right)$$

Use the distributive property to multiply 0.5 by 0.2304-0.192x+0.04x^{2}.

$$4704=0.4x^{2}+0.1152-0.096x+0.02x^{2}$$

Combine 0.4x^{2} and 0.02x^{2} to get 0.42x^{2}.

$$4704=0.42x^{2}+0.1152-0.096x$$

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

$$0.42x^{2}+0.1152-0.096x=4704$$

Subtract 0.1152 from both sides.

$$0.42x^{2}-0.096x=4704-0.1152$$

Subtract 0.1152 from 4704 to get 4703.8848.

$$0.42x^{2}-0.096x=4703.8848$$

Divide both sides of the equation by 0.42, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.42x^{2}-0.096x}{0.42}=\frac{4703.8848}{0.42}$$

Dividing by 0.42 undoes the multiplication by 0.42.

$$x^{2}+\left(-\frac{0.096}{0.42}\right)x=\frac{4703.8848}{0.42}$$

Divide -0.096 by 0.42 by multiplying -0.096 by the reciprocal of 0.42.

$$x^{2}-\frac{8}{35}x=\frac{4703.8848}{0.42}$$

Divide 4703.8848 by 0.42 by multiplying 4703.8848 by the reciprocal of 0.42.

$$x^{2}-\frac{8}{35}x=\frac{1959952}{175}$$

Divide -\frac{8}{35}, the coefficient of the x term, by 2 to get -\frac{4}{35}. Then add the square of -\frac{4}{35} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{35}x+\left(-\frac{4}{35}\right)^{2}=\frac{1959952}{175}+\left(-\frac{4}{35}\right)^{2}$$

Square -\frac{4}{35} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{35}x+\frac{16}{1225}=\frac{1959952}{175}+\frac{16}{1225}$$

Add \frac{1959952}{175} to \frac{16}{1225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{35}x+\frac{16}{1225}=\frac{2743936}{245}$$

Factor x^{2}-\frac{8}{35}x+\frac{16}{1225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{35}\right)^{2}=\frac{2743936}{245}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{35}\right)^{2}}=\sqrt{\frac{2743936}{245}}$$

Simplify.

$$x-\frac{4}{35}=\frac{8\sqrt{214370}}{35} x-\frac{4}{35}=-\frac{8\sqrt{214370}}{35}$$

Add \frac{4}{35} to both sides of the equation.

$$x=\frac{8\sqrt{214370}+4}{35} x=\frac{4-8\sqrt{214370}}{35}$$