Solve 64*left(0.3-xright)^2=2x^2

$$64 \times { \left(0.3-x \right) }^{ 2 } =2 { x }^{ 2 }$$

Solve for x

$$x=\frac{6\sqrt{2}+48}{155}\approx 0.36442117$$

$$x=\frac{48-6\sqrt{2}}{155}\approx 0.254933669$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.3-x\right)^{2}.

$$64\left(0.09-0.6x+x^{2}\right)=2x^{2}$$

Use the distributive property to multiply 64 by 0.09-0.6x+x^{2}.

$$5.76-38.4x+64x^{2}=2x^{2}$$

Subtract 2x^{2} from both sides.

$$5.76-38.4x+64x^{2}-2x^{2}=0$$

Combine 64x^{2} and -2x^{2} to get 62x^{2}.

$$5.76-38.4x+62x^{2}=0$$

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.

$$62x^{2}-38.4x+5.76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 62 for a, -38.4 for b, and 5.76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-38.4\right)±\sqrt{\left(-38.4\right)^{2}-4\times 62\times 5.76}}{2\times 62}$$

Square -38.4 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-38.4\right)±\sqrt{1474.56-4\times 62\times 5.76}}{2\times 62}$$

Multiply -4 times 62.

$$x=\frac{-\left(-38.4\right)±\sqrt{1474.56-248\times 5.76}}{2\times 62}$$

Multiply -248 times 5.76.

$$x=\frac{-\left(-38.4\right)±\sqrt{\frac{36864-35712}{25}}}{2\times 62}$$

Add 1474.56 to -1428.48 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-38.4\right)±\sqrt{46.08}}{2\times 62}$$

Take the square root of 46.08.

$$x=\frac{-\left(-38.4\right)±\frac{24\sqrt{2}}{5}}{2\times 62}$$

The opposite of -38.4 is 38.4.

$$x=\frac{38.4±\frac{24\sqrt{2}}{5}}{2\times 62}$$

Multiply 2 times 62.

$$x=\frac{38.4±\frac{24\sqrt{2}}{5}}{124}$$

Now solve the equation x=\frac{38.4±\frac{24\sqrt{2}}{5}}{124} when ± is plus. Add 38.4 to \frac{24\sqrt{2}}{5}.

$$x=\frac{24\sqrt{2}+192}{5\times 124}$$

Divide \frac{192+24\sqrt{2}}{5} by 124.

$$x=\frac{6\sqrt{2}+48}{155}$$

Now solve the equation x=\frac{38.4±\frac{24\sqrt{2}}{5}}{124} when ± is minus. Subtract \frac{24\sqrt{2}}{5} from 38.4.

$$x=\frac{192-24\sqrt{2}}{5\times 124}$$

Divide \frac{192-24\sqrt{2}}{5} by 124.

$$x=\frac{48-6\sqrt{2}}{155}$$

The equation is now solved.

$$x=\frac{6\sqrt{2}+48}{155} x=\frac{48-6\sqrt{2}}{155}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.3-x\right)^{2}.

$$64\left(0.09-0.6x+x^{2}\right)=2x^{2}$$

Use the distributive property to multiply 64 by 0.09-0.6x+x^{2}.

$$5.76-38.4x+64x^{2}=2x^{2}$$

Subtract 2x^{2} from both sides.

$$5.76-38.4x+64x^{2}-2x^{2}=0$$

Combine 64x^{2} and -2x^{2} to get 62x^{2}.

$$5.76-38.4x+62x^{2}=0$$

Subtract 5.76 from both sides. Anything subtracted from zero gives its negation.

$$-38.4x+62x^{2}=-5.76$$

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.

$$62x^{2}-38.4x=-5.76$$

Divide both sides by 62.

$$\frac{62x^{2}-38.4x}{62}=-\frac{5.76}{62}$$

Dividing by 62 undoes the multiplication by 62.

$$x^{2}+\left(-\frac{38.4}{62}\right)x=-\frac{5.76}{62}$$

Divide -38.4 by 62.

$$x^{2}-\frac{96}{155}x=-\frac{5.76}{62}$$

Divide -5.76 by 62.

$$x^{2}-\frac{96}{155}x=-\frac{72}{775}$$

Divide -\frac{96}{155}, the coefficient of the x term, by 2 to get -\frac{48}{155}. Then add the square of -\frac{48}{155} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{96}{155}x+\left(-\frac{48}{155}\right)^{2}=-\frac{72}{775}+\left(-\frac{48}{155}\right)^{2}$$

Square -\frac{48}{155} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{96}{155}x+\frac{2304}{24025}=-\frac{72}{775}+\frac{2304}{24025}$$

Add -\frac{72}{775} to \frac{2304}{24025} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{96}{155}x+\frac{2304}{24025}=\frac{72}{24025}$$

Factor x^{2}-\frac{96}{155}x+\frac{2304}{24025}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{48}{155}\right)^{2}=\frac{72}{24025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{48}{155}\right)^{2}}=\sqrt{\frac{72}{24025}}$$

Simplify.

$$x-\frac{48}{155}=\frac{6\sqrt{2}}{155} x-\frac{48}{155}=-\frac{6\sqrt{2}}{155}$$

Add \frac{48}{155} to both sides of the equation.

$$x=\frac{6\sqrt{2}+48}{155} x=\frac{48-6\sqrt{2}}{155}$$