Solve w^20.05^2+left(1-wright)^20.08^2+2wleft(1-wright)0.3*0.05*0.08=0.062169

$${ w }^{ 2 } { 0.05 }^{ 2 } + { \left(1-w \right) }^{ 2 } { 0.08 }^{ 2 } +2w \left( 1-w \right) 0.3 \cdot 0.05 \cdot 0.08=0.062169$$

Solve for w

$$w=\frac{\sqrt{3895385}}{650}+0.8\approx 3.836419957$$

$$w=-\frac{\sqrt{3895385}}{650}+0.8\approx -2.236419957$$

Steps Using the Quadratic Formula

Calculate 0.05 to the power of 2 and get 0.0025.

$$w^{2}\times 0.0025+\left(1-w\right)^{2}\times 0.08^{2}+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-w\right)^{2}.

$$w^{2}\times 0.0025+\left(1-2w+w^{2}\right)\times 0.08^{2}+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Calculate 0.08 to the power of 2 and get 0.0064.

$$w^{2}\times 0.0025+\left(1-2w+w^{2}\right)\times 0.0064+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Use the distributive property to multiply 1-2w+w^{2} by 0.0064.

$$w^{2}\times 0.0025+0.0064-0.0128w+0.0064w^{2}+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Combine w^{2}\times 0.0025 and 0.0064w^{2} to get 0.0089w^{2}.

$$0.0089w^{2}+0.0064-0.0128w+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Multiply 2 and 0.3 to get 0.6.

$$0.0089w^{2}+0.0064-0.0128w+0.6w\left(1-w\right)\times 0.05\times 0.08=0.062169$$

Multiply 0.6 and 0.05 to get 0.03.

$$0.0089w^{2}+0.0064-0.0128w+0.03w\left(1-w\right)\times 0.08=0.062169$$

Multiply 0.03 and 0.08 to get 0.0024.

$$0.0089w^{2}+0.0064-0.0128w+0.0024w\left(1-w\right)=0.062169$$

Use the distributive property to multiply 0.0024w by 1-w.

$$0.0089w^{2}+0.0064-0.0128w+0.0024w-0.0024w^{2}=0.062169$$

Combine -0.0128w and 0.0024w to get -0.0104w.

$$0.0089w^{2}+0.0064-0.0104w-0.0024w^{2}=0.062169$$

Combine 0.0089w^{2} and -0.0024w^{2} to get 0.0065w^{2}.

$$0.0065w^{2}+0.0064-0.0104w=0.062169$$

Subtract 0.062169 from both sides.

$$0.0065w^{2}+0.0064-0.0104w-0.062169=0$$

Subtract 0.062169 from 0.0064 to get -0.055769.

$$0.0065w^{2}-0.055769-0.0104w=0$$

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.

$$0.0065w^{2}-0.0104w-0.055769=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0065 for a, -0.0104 for b, and -0.055769 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-0.0104\right)±\sqrt{\left(-0.0104\right)^{2}-4\times 0.0065\left(-0.055769\right)}}{2\times 0.0065}$$

Square -0.0104 by squaring both the numerator and the denominator of the fraction.

$$w=\frac{-\left(-0.0104\right)±\sqrt{0.00010816-4\times 0.0065\left(-0.055769\right)}}{2\times 0.0065}$$

Multiply -4 times 0.0065.

$$w=\frac{-\left(-0.0104\right)±\sqrt{0.00010816-0.026\left(-0.055769\right)}}{2\times 0.0065}$$

Multiply -0.026 times -0.055769 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$w=\frac{-\left(-0.0104\right)±\sqrt{0.00010816+0.001449994}}{2\times 0.0065}$$

Add 0.00010816 to 0.001449994 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$w=\frac{-\left(-0.0104\right)±\sqrt{0.001558154}}{2\times 0.0065}$$

Take the square root of 0.001558154.

$$w=\frac{-\left(-0.0104\right)±\frac{\sqrt{3895385}}{50000}}{2\times 0.0065}$$

The opposite of -0.0104 is 0.0104.

$$w=\frac{0.0104±\frac{\sqrt{3895385}}{50000}}{2\times 0.0065}$$

Multiply 2 times 0.0065.

$$w=\frac{0.0104±\frac{\sqrt{3895385}}{50000}}{0.013}$$

Now solve the equation w=\frac{0.0104±\frac{\sqrt{3895385}}{50000}}{0.013} when ± is plus. Add 0.0104 to \frac{\sqrt{3895385}}{50000}.

$$w=\frac{\frac{\sqrt{3895385}}{50000}+\frac{13}{1250}}{0.013}$$

Divide \frac{13}{1250}+\frac{\sqrt{3895385}}{50000} by 0.013 by multiplying \frac{13}{1250}+\frac{\sqrt{3895385}}{50000} by the reciprocal of 0.013.

$$w=\frac{\sqrt{3895385}}{650}+\frac{4}{5}$$

Now solve the equation w=\frac{0.0104±\frac{\sqrt{3895385}}{50000}}{0.013} when ± is minus. Subtract \frac{\sqrt{3895385}}{50000} from 0.0104.

$$w=\frac{-\frac{\sqrt{3895385}}{50000}+\frac{13}{1250}}{0.013}$$

Divide \frac{13}{1250}-\frac{\sqrt{3895385}}{50000} by 0.013 by multiplying \frac{13}{1250}-\frac{\sqrt{3895385}}{50000} by the reciprocal of 0.013.

$$w=-\frac{\sqrt{3895385}}{650}+\frac{4}{5}$$

The equation is now solved.

$$w=\frac{\sqrt{3895385}}{650}+\frac{4}{5} w=-\frac{\sqrt{3895385}}{650}+\frac{4}{5}$$

Steps for Completing the Square

Calculate 0.05 to the power of 2 and get 0.0025.

$$w^{2}\times 0.0025+\left(1-w\right)^{2}\times 0.08^{2}+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(1-w\right)^{2}.

$$w^{2}\times 0.0025+\left(1-2w+w^{2}\right)\times 0.08^{2}+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Calculate 0.08 to the power of 2 and get 0.0064.

$$w^{2}\times 0.0025+\left(1-2w+w^{2}\right)\times 0.0064+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Use the distributive property to multiply 1-2w+w^{2} by 0.0064.

$$w^{2}\times 0.0025+0.0064-0.0128w+0.0064w^{2}+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Combine w^{2}\times 0.0025 and 0.0064w^{2} to get 0.0089w^{2}.

$$0.0089w^{2}+0.0064-0.0128w+2w\left(1-w\right)\times 0.3\times 0.05\times 0.08=0.062169$$

Multiply 2 and 0.3 to get 0.6.

$$0.0089w^{2}+0.0064-0.0128w+0.6w\left(1-w\right)\times 0.05\times 0.08=0.062169$$

Multiply 0.6 and 0.05 to get 0.03.

$$0.0089w^{2}+0.0064-0.0128w+0.03w\left(1-w\right)\times 0.08=0.062169$$

Multiply 0.03 and 0.08 to get 0.0024.

$$0.0089w^{2}+0.0064-0.0128w+0.0024w\left(1-w\right)=0.062169$$

Use the distributive property to multiply 0.0024w by 1-w.

$$0.0089w^{2}+0.0064-0.0128w+0.0024w-0.0024w^{2}=0.062169$$

Combine -0.0128w and 0.0024w to get -0.0104w.

$$0.0089w^{2}+0.0064-0.0104w-0.0024w^{2}=0.062169$$

Combine 0.0089w^{2} and -0.0024w^{2} to get 0.0065w^{2}.

$$0.0065w^{2}+0.0064-0.0104w=0.062169$$

Subtract 0.0064 from both sides.

$$0.0065w^{2}-0.0104w=0.062169-0.0064$$

Subtract 0.0064 from 0.062169 to get 0.055769.

$$0.0065w^{2}-0.0104w=0.055769$$

Divide both sides of the equation by 0.0065, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.0065w^{2}-0.0104w}{0.0065}=\frac{0.055769}{0.0065}$$

Dividing by 0.0065 undoes the multiplication by 0.0065.

$$w^{2}+\left(-\frac{0.0104}{0.0065}\right)w=\frac{0.055769}{0.0065}$$

Divide -0.0104 by 0.0065 by multiplying -0.0104 by the reciprocal of 0.0065.

$$w^{2}-1.6w=\frac{0.055769}{0.0065}$$

Divide 0.055769 by 0.0065 by multiplying 0.055769 by the reciprocal of 0.0065.

$$w^{2}-1.6w=\frac{55769}{6500}$$

Divide -1.6, the coefficient of the x term, by 2 to get -0.8. Then add the square of -0.8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}-1.6w+\left(-0.8\right)^{2}=\frac{55769}{6500}+\left(-0.8\right)^{2}$$

Square -0.8 by squaring both the numerator and the denominator of the fraction.

$$w^{2}-1.6w+0.64=\frac{55769}{6500}+0.64$$

Add \frac{55769}{6500} to 0.64 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$w^{2}-1.6w+0.64=\frac{59929}{6500}$$

Factor w^{2}-1.6w+0.64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w-0.8\right)^{2}=\frac{59929}{6500}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w-0.8\right)^{2}}=\sqrt{\frac{59929}{6500}}$$

Simplify.

$$w-0.8=\frac{\sqrt{3895385}}{650} w-0.8=-\frac{\sqrt{3895385}}{650}$$

Add 0.8 to both sides of the equation.

$$w=\frac{\sqrt{3895385}}{650}+\frac{4}{5} w=-\frac{\sqrt{3895385}}{650}+\frac{4}{5}$$